Deformed GOE

# Deformed GOE

## 1 Introduction

The work of B.L. is partially supported by NSERC. The work of H.-T. Y. is partially supported by NSF Grant DMS-1307444, DMS-1606305 and a Simons Investigator award.

In the pioneering work [74], Wigner introduced what are known as the Wigner random matrix ensembles. These ensembles consist of N\times N real symmetric (\beta=1) or complex Hermitian (\beta=2) random matrices W=(w_{ij}) whose entries are centered and independent (up to the symmetry constraint W=W^{*}) with variance

If the w_{ij}’s are independent real (resp., complex) Gaussians then the ensemble is called the Gaussian Orthogonal Ensemble (resp., Gaussian Unitary Ensemble) (GOE/GUE). Wigner conjectured that in the limit N\to\infty the local eigenvalue statistics are universal in that they depend only on the symmetry class of the matrix ensemble (real symmetric or complex Hermitian) and are otherwise independent of the underlying distribution of the matrix entries. After Wigner’s seminal work, Gaudin, Dyson and Mehta explicitly calculated the eigenvalue correlation functions in the Gaussian cases.

Mehta formalized the universality conjecture in the book [60] and stated that the correlation functions of general Wigner matrices should coincide with the GOE/GUE in the limit N\to\infty. There are several possible topologies in which this convergence could hold. Perhaps the most natural topology to consider is pointwise convergence of the correlation functions. However, this cannot hold for random matrix ensembles with discrete entries. One suitable topology is that of vague convergence of the correlation functions around an energy E, which we will call fixed energy universality. A weaker topology can be constructed by averaging over energies in a small window near E and asking for vague convergence of the energy-averaged quantities. We will call this averaged or unfixed energy universality. Finally, one can also ask for the vague convergence of the eigenvalue gaps with a fixed label (i.e., vague convergence of the random variable \lambda_{N/2+1}-\lambda_{N/2}) which we call gap universality.

There has recently been spectacular progress in proving the Wigner-Dyson-Mehta conjecture for a wide variety of random matrix ensembles. Bulk universality for Wigner matrices of all symmetry classes was proven in the works [34, 33, 35, 37, 40, 42]. Parallel results were established in certain cases in [68, 69], with the key result being a “four moment comparison theorem.” In this paper we are interested in the robust three-step approach to universality formulated and developed in the works [34, 33, 35, 37, 40, 42]. This approach consists of:

1. A high probability estimate of the eigenvalue density down to the almost-optimal scale \eta\sim N^{\varepsilon}/N. This establishes eigenvalue rigidity; that is, the bulk eigenvalues are close to their expectations

 |\lambda_{i}-\mathbb{E}[\lambda_{i}]|\varleq\frac{N^{\varepsilon}}{N} (1.2)

with overwhelming probability. Moreover, the expectations are determined by the quantiles of the macroscopic eigenvalue density.

2. Proving bulk universality for random matrix ensembles with a small additive Gaussian component. This is usually established by studying the rate of convergence of Dyson Brownian motion to local equilibrium.

3. A comparison or stability argument comparing a given random matrix ensemble to one with a small Gaussian component.

For complex Hermitian ensembles, Step 2 can be established by using an explicit algebraic formula, the Brézin-Hikami formula, to analyze the correlation functions. This idea was used by Johansson [49] and Ben Arous-Peche [10] who established bulk universality for ensembles with an order 1 Gaussian component, i.e., establishing that the time to equilibrium is at most order 1 in this case. The optimal time to equilibrium in the second step, i.e., for t\gtrsim 1/N, was established in [35] where the Brézin-Hikami formula and estimates from the local semicircle law were the key tools. In this special algebraic case, the second step yields fixed energy universality and so the WDM conjecture was established for complex Hermitian matrices in this strong sense [35, 68, 36].

An analogue of the Brézin-Hikami formula is unknown in the real symmetric case and therefore the approach [35] could not be extended to real symmetric Wigner ensembles. A new approach based on the local relaxation flow of Dyson Brownian motion (DBM) was developed in the works [38, 37, 41]. DBM is defined by applying an independent (up to the symmetry constraint H=H^{*}) Ornstein-Uhlenbeck process to every matrix element; Dyson computed the flow on the eigenvalues and found that they satisfy a closed system of stochastic differential equations. The approach of [38, 37, 41] is based on this representation and applies to all symmetry classes as well as sample covariance matrices and sparse ensembles. However, it yields only averaged energy universality, albeit with the averaging taken over a very small window.

The second step was finally completed for real symmetric Wigner ensembles in the sense of fixed energy in [17]. By developing a sophisticated homogenization theory for a discrete parabolic equation derived from DBM, the authors proved that after a time t=o(1), the local statistics of Dyson Brownian motion started from a Wigner ensemble coincide with that of the GOE in the fixed energy sense. As the third step in the three-step strategy described above is insensitive to the mode of convergence of the correlation functions, this proved the Wigner-Dyson-Mehta conjecture in the fixed energy sense for all symmetry classes.

The time to equilibrium proven in the work [17] is relatively long, t\sim N^{-\varepsilon} for a small \varepsilon>0, which moreover depends on the choice of test function. This limits the applicability of the work [17] in proving fixed energy universality for other ensembles. For example, it does not imply fixed energy universality for sparse random graphs, for which averaged energy universality is known [47, 1, 33, 34]. Moreover, the approach relies on the fact that the global eigenvalue density of the initial data is given by the semicircle law.

The analysis of DBM developed in the works [38, 37, 41] is in some sense global as it relies on the fact that DBM with initial data a Wigner matrix will follow the semicircle law. In the work [40] the correlation functions were expressed as time averages of random walks in a random environment. This allows for a local analysis of the dynamics and various tools from PDE (such as Hölder regularity via the di-Giorgi-Nash-Moser method) and stochastic analysis can be applied.

In the work [55] the time to equilibrium of DBM for a wide class of initial data (going beyond the Wigner class) was studied (see also [39] for related results). For random matrix ensembles that have a local density down to scales \eta\gtrsim 1/N, it was proven that the time to local equilibrium is t\gtrsim 1/N, in the sense of both averaged energy and gap universality.

There have been several recent works extending the Wigner-Dyson-Mehta conjecture beyond the class of Wigner matrices, such as to sparse random graphs [45, 47, 8, 7, 33, 34, 1], matrices with correlated entries [23, 6, 2], deformed Wigner ensembles [56, 57], certain classes of band matrices [13] and the general Wigner-type matrices of [3, 5, 4]. These works generally follow the three-step strategy outlined above. In many of these cases, the works [55, 39] essentially complete the second step of this approach. As the results [55, 39] imply averaged energy universality, any work relying on [55, 39] for the second step establishes the Wigner-Dyson-Mehta conjecture in only the averaged energy sense.

In the current work we establish that the time to local equilibrium for DBM is t\gtrsim N^{-1} for a wide class of initial data, in the fixed energy sense. The main assumption on the initial data is that the density of states is bounded above and below down to scales \eta\ll t in a window of size L\gg\sqrt{t}. As a consequence, fixed energy universality is established for essentially all random matrix ensembles for which previously only averaged energy universality could be proven.

One of the key insights of [17] is that the difference of two coupled DBM flows obeys a discrete nonlocal parabolic equation. One of the main results of [17] is a homogenization theory for this parabolic equation. This theory shows that the solution of the discrete parabolic equation is given by the discretization of the continuum limit, and this reduces the problem of microscopic statistics to an easier mesoscopic problem.

Our approach follows the same high-level strategy in that we couple two DBM flows and develop a homogenization theory for the resulting parabolic equation. The generator of the parabolic equation of [17] is hard to control. To deal with this we modify the coupling of [17] and introduce a continuous interpolation. This gives us a family of parabolic equations whose generators have better properties.

The homogenization theory of [17] was based around a Duhamel expansion and estimating the coefficients of the generator. The short range part of the generator is quite singular and was controlled using an energy estimate and the discrete Di-Giorgi-Nash-Moser theorem of [40]. This caused some restriction on the time to equilibrium that could be proven.

Our method is based around the standard \ell^{2}-energy method and a discrete Sobolev inequality. The energy method gives us an estimate on the time average of the discrete \dot{H}^{1/2} norm of the difference between the fundamental solution of the discrete equation and its continuum limit. This allows us to get a time-averaged \ell^{\infty} estimate on the fundamental solution of the discrete parabolic equation via a discrete Sobolev inequality. We then use the semigroup property to remove the time average.

In order to carry this out one needs a good ansatz for comparison with the discrete fundamental solution. We substitute the particle location coming from the DBM into the fundamental solution of the continuum limit. With this approach a martingale term, as well as other errors of lower order, arises in the energy method, but we are able to control them using heat kernel bounds for our process. This ansatz first appeared in [18] and has been used independently in [12] to analyze extremal gap statistics of Wigner ensembles.

We find that the limiting hydrodynamic equation is a fairly simple nonlocal parabolic equation describing a symmetric jump process on \mathbb{R}. The heat kernels of such processes have been studied recently in, e.g., [25, 24] and we partially rely on their work in our analysis of the limiting equation.

Our homogenization theory has an advantage over [17] in that our estimates hold with overwhelming probability (i.e., \vargeq 1-N^{-D} for any large D>0). The homogenization theory [17] relied on certain level repulsion estimates and as a consequence the main estimates were only known to hold with polynomially high probability (i.e., \vargeq 1-N^{-\varepsilon} for some small \varepsilon>0). While this is not significant to the application of universality, we believe that this improvement is important for future applications. For example, if one wishes to study the maximal eigenvalue gap in the bulk of generalized Wigner matrices, then one can use the homogenization result given here together with a union bound over order N eigenvalues. This approach has been carried out in the work [54]. Note that the result of [17] would not be sufficient to study this spectral statistic.

Moreover our method is robust in that it essentially relies only on rigidity; in many random matrix ensembles optimal level repulsion estimates (on which the previous methods [17, 55, 39] relied) are not known and can be hard to establish.

As mentioned above, the homogenization theory reduces the microscopic problem of fixed energy universality to a problem involving linear mesoscopic statistics. Central limit theorems for mesoscopic linear statistics of Wigner matrices were established first in certain scales in [21, 22, 59] and then down to the almost-optimal scale \eta=N^{\varepsilon}/N in [44]. Mesoscopic statistics of compactly supported test functions for the special case of \beta=2 for DBM with deterministic initial data was established in [31]. The analysis in [31] relied on the Brézin-Hikami formula special to the \beta=2 case and cannot be applied here. Moreover the test function coming from the homogenization theorem is not of compact support - only its derivative is - and has no spatial decay, which presents a serious complication. The mesoscopic results [21, 22, 59, 44, 31] all apply only to functions with either compact support or at least some spatial decay as |x|\to\infty.

In the present work we establish a mesoscopic central limit theorem for DBM for a certain class of non-compactly supported test functions which have no spatial decay. As an aside, we remark that if our methods are restricted to the compactly supported case, then we can prove that if the scale of the function is less than t, then the linear statistic coincides with the GOE. Here, in the compactly supported case one can remove the restrictions that we encounter in the non-compactly supported case. This is an extension of some of the results of [31] to \beta=1. Our main interest in a mesoscopic central limit theorem is to analyze the statistic coming from the homogenization theory, and so we only settles for a few remarks concerning test functions of compact support - see Section 6.

The works [55, 39] establishing averaged energy universality for DBM relied heavily on the discrete Di-Giorgi-Nash-Moser theorem of [40]. As a consequence, the rate of convergence was somewhat non-explicit. While in this work we do not attempt to derive optimal error bounds, our result improves on [55, 39] in the sense that our bounds can be quantified explicitly in terms of the parameters of the model.

### 1.1 Applications

Our homogenization theory also allows us to establish universality of the space-time DBM process — that is, up to an explicit deterministic shift in space and a re-scaling in space and time, the multitime correlation functions of DBM coincide with the GOE/GUE.

Averaged energy universality for general one-cut \beta-ensembles was established first in [14, 16, 15, 40]. Further results for bulk universality for multi-cut potentials were established in [65, 9]. Fixed energy universality for one-cut C^{4} potentials was announced in [32] and can be proven using the methods of [17]. Previously, Shcherbina had established fixed energy universality for analytic potentials in the multi-cut case in [65]. For completeness we sketch how our methods can be adapted to re-prove the results of [32, 17] and moreover establish a polynomial error estimate.

#### 1.1.1 Fixed energy universality for general random matrix ensembles

Many of the recent works on universality of general random matrix ensembles have relied on the aforementioned three-step strategy to proving universality. In the second step these works relied on [55, 39] for universality for the Gaussian divisible ensembles. The works [55, 39] provided both gap universality and averaged energy universality for the Gaussian divisible ensembles; consequently this form of universality has been proven, for example, for the adjacency matrices of sparse random graphs [45, 47, 8, 7, 33, 34, 1], matrices with correlated entries [23, 6, 2] and the general Wigner-type matrices of [3, 5, 4]. By instead relying on the current work, fixed energy universality is established for all of these ensembles.

#### 1.1.2 Eigenvalue interval probabilities

Fixed energy universality has several other consequences which we now outline. It establishes the existence of the local density of states on microscopic scales as well as the universality of the Jimbo-Miwa-Mori-Sato formula for the gap probability. In addition, it implies universality of the distribution of the smallest singular value of various random matrix ensembles, including the adjacency matrices of a wide variety of sparse random graphs which is of interest in computer science.

#### 1.1.3 Invertibility of symmetric random matrices

The invertibility problem in random matrix theory is typically divided into two components [71]. The first is whether a random matrix is invertible with high probability, and the second is to determine the typical size of the norm of the inverse, or size of the smallest singular value. A motivating problem of the former type is the conjecture that an iid Bernoulli matrix is singular with probability less than (2+o(1))^{-N}. Komlos [52] first proved that the singularity probability is vanishing. An exponential bound was first obtained in [50] and later improved in [66, 20].

The size of the inverse is related to the condition number which plays a crucial role in applied linear algebra. For example, the condition number controls the complexity or numerical accuracy in solving the linear equation Ax=b. Von Neumann and his collaborators speculated [72] that the least singular value satisfies s_{\mathrm{min}}(A)\sim N^{-1} for matrices with iid entries.

By now a large literature has emerged on the invertibility problem for both symmetric and iid ensembles. We refer to the surveys [63, 73, 61] and the references therein, and mention only a few specific results placing the present work in context. The invertibility of dense Erdős-Rényi graphs was established in [27] and was later extended to the sparse regime in [28]. The first estimate of the form s_{\mathrm{min}}(H)\sim N^{-1} for symmetric matrices was first obtained in [71]. In the iid case, it is even known that the distribution of the (properly rescaled) smallest singular value at the hard edge is universal [67]. However, in the sparse regime little is known about the size of the smallest singular value in the symmetric case.

It is an open conjecture that the adjacency matrix of a random d-regular graph is invertible with high probability for d\vargeq 3 [28, 73, 43]. This problem is of interest in universal packet recovery [30]. In the case of random d-regular directed graphs, substantial progress has been made by [26, 58]. It is also conjectured that the adjacency matrices of more general sparse graphs outside the Erdos-Renyi class should be invertible with high probability [28].

The invertibility of many classes of random matrices is in fact a corollary of previous works by two of the current authors [55, 47] as well as others [1, 23, 8, 7, 3, 6]. These classes include, for example, adjacency matices of random regular graphs, matrices with correlated entries and general sparse random matrices. To fix ideas we consider the d-regular random graph with adjacency matrix A, where

 d\vargeq N^{\varepsilon}. (1.3)

The invertibility of A+G where G is a small GOE component follows from Section 5 of [55] as well as the local law of [8]. The comparison methods of [47, 7] then allow the invertibility to be transferred back to A. This proves the conjecture of [28, 43, 73] in the regime (1.3). This methodology extends to the other random matrix ensembles considered in [47], as well as those considered in [1, 23, 3, 6] as long as 0 lies in the bulk of the spectrum.

This strategy yields an additional effective estimate on the size of the inverse in all cases, which was previously known only in the non-sparse regime. That is, there is a c>0 so that for all sufficiently small \varepsilon>0,

 \mathbb{P}[||A^{-1}||\vargeq N^{\varepsilon}/N]\varleq C(\varepsilon)N^{-c% \varepsilon}. (1.4)

For example, one can take A to be the (properly rescaled so that the spectrum lies in a window of order 1) adjacency matrix of a sparse d-regular or Erdős-Rényi graph.

The current work goes beyond this and establishes universality of the smallest singular value of many random matrix ensembles. Our work implies, for example, that for each t\in\mathbb{R},

 \lim_{N\to\infty}\left|\mathbb{P}[||A^{-1}||\vargeq tN]-\mathbb{P}[||H^{-1}||% \vargeq tN]\right|=0, (1.5)

where A is the (again, properly rescaled) adjacency matrix of a random d-regular graph and H is a GOE matrix (again for d in the regime (1.3)).

### 1.2 Overview

The remainder of the paper is as follows. In Section 2 we introduce precisely our model and state our main results, applications and auxilliary results. Section 3 contains the main part of the homogenization results and Section 4 contains the proofs of certain a-priori bounds on the heat kernel. We study and prove regularity of the limiting continuum equation in Section 5. In Section 6 we prove our results on mesoscopic linear statistics for DBM. Section 7 contains the proof of fixed energy universality using the homogenization theory and the central limit theorem for mesoscopic linear statistics. In Section 8 we sketch the proof of fixed energy universality for \beta-ensembles.

Acknowledgements. B.L. thanks Jiaoyang Huang for useful discussions.

## 2 Model

Let V be a deterministic diagonal matrix and let W be a standard GOE matrix. We consider the following model

 H_{t}=V+\sqrt{t}W. (2.1)

We make the following assumptions on V.

###### Definition 2.1.

Let G=G_{N} and g=g_{N} be N-dependent parameters. For definiteness we assume that there is a \delta>0 s.t. N^{-\delta}\vargeq g\vargeq N^{\delta}/N and G\varleq N^{-\delta}. This \delta will not be important in the method or the main results. We say that V is (g,G)-regular if

 c\varleq\mathrm{Im}\mbox{ }[m_{V}(E+\mathrm{i}\eta)]\varleq C (2.2)

for |E|\varleq G and g\varleq\eta\varleq 10, and if there is a C_{V}>0 s.t.

 ||V||\varleq N^{C_{V}}. (2.3)

Remark. The assumption (2.3) is technical and can be removed with some minor work. We omit this from the current paper.

We will be considering times satisfying gN^{\sigma}\varleq t\varleq N^{-\sigma}G^{2}. We also introduce here the frequently used notation z=E+\mathrm{i}\eta for E,\eta\in\mathbb{R}.

### 2.1 Free convolution

In this section we introduce the free convolution. The semicircle law is given by

 \rho_{\mathrm{sc}}(E):=\frac{1}{2\pi}\boldsymbol{1}_{\{|E|\varleq 2\}}\sqrt{4-% E^{2}}. (2.4)

It describes the limiting eigenvalue density of the GOE. The eigenvalue density of H_{t} does not follow the semicircle law and is given by a free convolution. We define the free convolution of V with the semicircle law at time t via its Stieltjes transform which we denote by m_{\mathrm{fc},t}. The function m_{\mathrm{fc},t} is defined as the unique solution to

The free convolution law is defined by

 \rho_{\mathrm{fc},t}(E):=\lim_{\eta\downarrow 0}\frac{1}{\pi}\mathrm{Im}\mbox{% }[m_{\mathrm{fc},t}(E+\mathrm{i}\eta)]. (2.6)

The free convolution is well-studied. For example, it is known that a unique solution to (2.5) exists and that \rho_{\mathrm{fc},t} is analytic on the interior of its support. We refer to [11] for further details. We will also denote the free convolution law at time t by \rho_{\mathrm{fc},t}(E):=\rho_{V}\boxplus\rho_{\mathrm{sc},t}.

### 2.2 Fixed energy universality

Let p_{H_{t}}^{(N)} denote the symmetrized eigenvalue density of H_{t}. The k-point correlation functions are defined by

 p_{H_{t}}^{(k)}(\lambda_{1},\cdots,\lambda_{k}):=\int p_{H_{t}}^{(N)}(\lambda_% {1},\cdots,\lambda_{N})\mathrm{d}\lambda_{k+1}\cdots\mathrm{d}\lambda_{N}. (2.7)

The corresponding objects for the GOE are denoted p^{(N)}_{GOE} and p^{(k)}_{GOE}. The following is our main result which states that the k-point correlation functions of H_{t} converges to those of the GOE in the fixed energy sense.

###### Theorem 2.2.

Let V be a deterministic (g,G)-regular diagonal matrix. Let \sigma>0 and let

 gN^{\sigma}\varleq t\varleq N^{-\sigma}G^{2}. (2.8)

Let 0<q<1 and let |E|\varleq qG. There is a constant \kappa>0 so that the following holds. For every k and smooth test function O\in C^{\infty}_{c}(\mathbb{R}^{k}) there is a constant C>0 such that

 \displaystyle\bigg{|} \displaystyle\int O(\alpha_{1},...,\alpha_{k})p_{H_{t}}^{(k)}\left(E+\frac{% \alpha_{1}}{N\rho_{\mathrm{fc},t}(E)},\cdots E+\frac{\alpha_{k}}{N\rho_{% \mathrm{fc},t}(E)}\right)\mathrm{d}\alpha_{1}\cdots\mathrm{d}\alpha_{k} \displaystyle- \displaystyle\int O(\alpha_{1},...,\alpha_{k})p_{GOE}^{(k)}\left(E+\frac{% \alpha_{1}}{N\rho_{\mathrm{sc}}(E)},\cdots E+\frac{\alpha_{k}}{N\rho_{\mathrm{% sc}}(E)}\right)\mathrm{d}\alpha_{1}\cdots\mathrm{d}\alpha_{k}\bigg{|}\varleq CN% ^{-\kappa} (2.9)

#### 2.2.1 Applications to other ensembles

Theorem 2.2 implies fixed energy universality for a wide variety of ensembles appearing in random matrix theory. Recall the three-step strategy to proving universality for random matrix ensembles outlined in the introduction. Many recent works in random matrix theory used the results of [55, 39] to complete the second step. The input of [55, 39] is to provide universality of DBM started from the chosen random matrix ensemble in either the fixed gap sense or averaged energy sense. The third step is relatively insensitive to the type of universality proven in the second step. Therefore, if one uses Theorem 2.2 instead of [55, 39] one can prove fixed energy universality for following ensembles.

1. Sparse random matrix ensembles such as the adjacency matrices of random graphs [47, 8, 7, 34, 33, 1, 45]

2. The general Wigner-type matrices of [3, 5, 4].

3. Matrices with correlated entries [23, 6, 2].

4. Deformed Wigner ensembles [56, 57].

Lastly, while fixed energy universality of generalized Wigner matrices was settled in [17], our methods yield a polynomial rate of convergence which was previously unknown.

### 2.3 Further results

#### 2.3.1 Multitime correlation functions

The eigenvalues \lambda_{i} of H_{t} at each fixed time are equal in distribution to the unique strong solution of the system of the following system of SDEs, known as Dyson Brownian motion:

 \mathrm{d}\lambda_{i}=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum_{j}% \frac{1}{\lambda_{i}-\lambda_{j}}\mathrm{d}t (2.10)

with initial data \lambda_{i}(0)=V_{i}. Theorem 2.2 implies that at each fixed time, the correlation functions of \{\lambda_{i}(t)\}_{i} coincide with the GOE. Our methods also allow us to consider multitime correlation functions. For simplicity we just state the result for two times t_{a}<t_{b}. One can also consider any finite set of times t_{a},...,t_{k}. Given two times t_{a}<t_{b} let p_{t_{a},t_{b}}(\lambda_{1}(t_{a}),\cdots\lambda_{N}(t_{a}),\lambda_{1}(t_{b})% ,\cdots\lambda_{N}(t_{b})) denote the symmetrized density of \{\lambda_{i}(t_{a}),\lambda_{j}(t_{b})\}_{i,j}. The multitime k-point correlation function is defined by

 \displaystyle p^{(k)}_{t_{a},t_{b}}(\lambda_{1}(t_{a}),\cdots,\lambda_{k}(t_{a% }),\lambda_{1}(t_{b}),\cdots,\lambda_{k}(t_{b})) \displaystyle:= \displaystyle\int p_{t_{a},t_{b}}(\lambda_{1}(t_{a}),\cdots\lambda_{N}(t_{a}),% \lambda_{1}(t_{b}),\cdots\lambda_{N}(t_{b}))\mathrm{d}\lambda_{k+1}(t_{a})% \cdots\mathrm{d}\lambda_{N}(t_{a})\mathrm{d}\lambda_{k+1}(t_{b})\cdots\mathrm{% d}\lambda_{N}(t_{b}). (2.11)

Denote the analogous object for the GOE by p^{(k)}_{t_{a},t_{b},GOE} (i.e., start the process \lambda_{i} from the GOE ensemble). Fix an energy |E(t_{a})|\varleq qG and define E(t) for t>t_{a} by

 \partial_{t}E=\mathrm{Re}[m_{\mathrm{fc},t}(E)] (2.12)
###### Theorem 2.3.

Let V be as above and let gN^{\sigma}\varleq t_{a}\varleq N^{-\sigma}G^{2}. Let O be a smooth compactly supported test function. There is a constant \kappa>0 so that for any t_{a}\varleq t_{b}\varleq N^{\kappa}/N we have

 \displaystyle\bigg{|} \displaystyle\int O(\alpha_{1},\cdots\alpha_{k},\beta_{1},\cdots\beta_{k})p_{t% _{a},t_{b}}^{(k)}\left(E(t_{a})+\frac{\vec{\alpha}}{N\rho_{\mathrm{fc},t_{a}}(% E(t_{a}))},E(t_{b})+\frac{\vec{\beta}}{N\rho_{\mathrm{fc},t_{b}}(E(t_{b}))}% \right)\mathrm{d}\vec{\alpha}\mathrm{d}\vec{\beta} \displaystyle-\int O(\alpha_{1},\cdots\alpha_{k},\beta_{1},\cdots\beta_{k})p_{% t_{a},c_{t}t_{b},GOE}^{(k)}\left(E^{\prime}+\frac{\vec{\alpha}}{N\rho_{\mathrm% {sc}}(E^{\prime})},E^{\prime}+\frac{\vec{\beta}}{N\rho_{\mathrm{sc}}(E^{\prime% })}\right)\mathrm{d}\vec{\alpha}\mathrm{d}\vec{\beta}\bigg{|}\varleq N^{-% \kappa}, (2.13)

for any fixed E^{\prime}\in(-2,2). Above the constant c_{t} is defined by c_{t}:=(\rho_{\mathrm{sc}}(0)/\rho_{\mathrm{fc},t_{b}}(E(t_{b})))^{2}.

Remark. One can replace \rho_{\mathrm{fc},t_{b}}(E(t_{b})) by \rho_{\mathrm{fc},t_{a}}(E(t_{a})) as the difference is o(1).

#### 2.3.2 Jimbo-Miwa-Mori-Sato formula

Once one establishes fixed energy universality for a random matrix ensemble, it is a standard argument to determine the distribution of the number of eigenvalues in a interval of size c/N. More precisely, Theorem 2.2 implies that for intervals I_{1},\cdots,I_{k} and integers n_{1},\cdots n_{k} the probability

 \mathbb{P}\left[\left|\left\{\lambda_{i}\in E+\frac{I_{j}}{N\rho(E)}\right\}% \right|=n_{j},1\varleq j\varleq k\right] (2.14)

converges to that of the GOE where \rho(E) is the eigenvalue density of the ensemble under consideration.

For example, for the adjacency matrices of a class of sparse random graphs, we have

 \lim_{N\to\infty}\mathbb{P}\left[\left|\left\{\lambda_{i}\in\frac{[0,t]}{N\pi% \rho_{\mathrm{sc}}(0)}\right\}\right|=0\right]=E_{1}(0,t) (2.15)

where E_{1} is an explicit function of a solution to Painlevé equation.

#### 2.3.3 Invertibility of symmetric random matrices

The result (2.15) provides explicit information on the distribution of the size of the inverse of various random matrix ensembles. For example, (2.15) implies that for the adjacency matrices A of sparse Erdős-Rényi and d-regular graphs we have for every t>0,

 \lim_{N\to\infty}\left|\mathbb{P}\left[||A^{-1}||\vargeq Nt\right]-\mathbb{P}% \left[||H^{-1}||\vargeq Nt\right]\right|=0 (2.16)

where H is a GOE matrix. From previous results in the literature [47, 55, 7, 8] it is easily deduced that the adjacency matrix of a sparse Erdős-Rényi or d-regular graph is invertible with high probability. The result (2.16) is finer, in that it demonstrates that the limiting distribution of the size of the inverse, or equivalently, the size of the smallest singular value of A, is universal.

#### 2.3.4 Fixed energy universality for \beta-ensembles

Our methods also imply fixed energy universality for a class of \beta-ensembles. A \beta-ensemble is a measure on the simplex \lambda_{1}\varleq\cdots\varleq\lambda_{N} with probability density proportional to

 \mathrm{e}^{-\beta N\sum_{k}\frac{1}{2}V(\lambda_{k})+\beta\sum_{i

We assume that V is a C^{4} real function with second derivative bounded below and growth condition

 V(x)>(2+\alpha)\log(1+|x|) (2.18)

for all large x and an \alpha>0. The averaged density of the empirical spectral measure converges weakly to a continuous function \rho_{V}, the equilibrium density with compact support. We assume that \rho_{V} is supported on a single interval [A,B] and that V is regular in the sense of [53]. We denote the k-point correlation functions by p^{(k)}_{V} and those for the Gaussian \beta-ensemble (for V(x)=x^{2}/2) by p^{(k)}_{\beta}.

Under these conditions fixed energy universality was announced in [32] and can be proven using the methods of [17]. Previously, M. Shcherbina established fixed energy universality for multi-cut analytic \beta-ensembles in [65]. The following result is an improved version of the result in [32] in that it provides an error estimate N^{-\kappa} to the fixed energy universality. Similarly to [32], our methodology is based on the homogenization idea initiated in [32, 17].

###### Theorem 2.4.

Let V be as above and assume \beta\vargeq 1. Let E\in(A,B) and E^{\prime}\in(-2,2). Let O be a smooth test function. There is a \kappa>0 such that

 \displaystyle\bigg{|} \displaystyle\int O(\alpha_{1},...,\alpha_{k})p_{V}^{(k)}\left(E+\frac{\alpha_% {1}}{N\rho_{V}(E)},\cdots E+\frac{\alpha_{k}}{N\rho_{V}(E)}\right)\mathrm{d}% \alpha_{1}\cdots\mathrm{d}\alpha_{k} \displaystyle- \displaystyle\int O(\alpha_{1},...,\alpha_{k})p_{\beta}^{(k)}\left(E^{\prime}+% \frac{\alpha_{1}}{N\rho_{\mathrm{sc}}(E^{\prime})},\cdots E^{\prime}+\frac{% \alpha_{k}}{N\rho_{\mathrm{sc}}(E^{\prime})}\right)\mathrm{d}\alpha_{1}\cdots% \mathrm{d}\alpha_{k}\bigg{|}\varleq CN^{-\kappa} (2.19)

Remark. It is also possible to deduce analogous results for multitime correlation functions in the following sense. If one modifies (2.10) to

 \mathrm{d}\lambda_{i}=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum_{j}% \frac{1}{\lambda_{i}-\lambda_{j}}\mathrm{d}t-\frac{V^{\prime}(\lambda_{i})}{2}% \mathrm{d}t (2.20)

then the \beta-ensemble with potential V is left invariant by this flow. One can prove that the multitime correlation functions coincide with the process (2.10) started from a Gaussian \beta-ensemble.

#### 2.3.5 Mesoscopic statistics for DBM

Our methodology of proving fixed energy universality reduces the microscopic problem to a problem involving mesoscopic linear statistics. In order to complete the proof of fixed energy universality we are forced to calculate mesoscopic statistics for DBM. Mesoscopic statistics have received some attention in the literature recently and we therefore state our result as it may be of independent interest.

###### Theorem 2.5.

Let \varphi be a smooth test function satisfying

where t^{\prime}=N^{\alpha}/N. Let V be (g,G)-regular and let |E|\varleq qG. Let t=N^{\omega}/N satisfy gN^{\sigma}\varleq t\varleq G^{2}N^{-\sigma}. Assume that \alpha>0 satisfies \omega/2<\alpha<\omega. Then the mesoscopic statistic

 \sum_{i}\varphi(\lambda_{i}) (2.22)

converges weakly to a Gaussian. If \varphi is not compactly supported, then the variance is bounded below by c|\log(t^{\prime}/t)|.

Remark. Our results are more general — see Section 6. We calculate the characteristic function with an explicit rate of convergence in a growing neighborhood of the origin.

If \varphi is compactly supported, then with some modifications of our methods one can remove the unnatural restriction \alpha>\omega/2. As the above theorem will suffice in our application to fixed energy universality we do not provide the details.

### 2.4 Local law and rigidity

In this section we recall the local law for H_{t}. These a-priori estimates are the key technical input of our methods. For times of order 1 the local law was established in [56, 57]. The argument was adapted to short times in [55]. The empirical Stieltjes transform of H_{t} will be denoted by

 m_{N}(z):=\frac{1}{N}\sum_{i}\frac{1}{\lambda_{i}-z}. (2.23)

Under the above hypotheses we have the following rigidity and local law estimates. We need some notation. For any 0<q<1 let

 \mathcal{I}_{q}:=[-qG,qG]. (2.24)

Let \varepsilon>0 and 0<q<1. We consider the spectral domain

 \displaystyle\mathcal{D}_{\varepsilon,q} \displaystyle=\left\{z=E+\mathrm{i}\eta:E\in\mathcal{I}_{q},N^{10C_{V}}\vargeq% \eta\vargeq N^{\varepsilon}/N\right\} \displaystyle\cup\left\{z:D+\mathrm{i}\eta:|E|\varleq N^{10C_{V}},N^{10C_{V}}% \vargeq\eta\vargeq c\right\} (2.25)

We have

###### Theorem 2.6.

Fix \varepsilon>0 and 0<q<1. Let \sigma>0 be such that gN^{\sigma}\varleq t\varleq N^{-\sigma}G^{2}. For any D>0 and \delta>0 we have

 \mathbb{P}\left[\sup_{z\in D_{\varepsilon}}\left|m_{N}(z)-m_{\mathrm{fc},t}(z)% \right|\vargeq\frac{N^{\delta}}{N\eta}\right]\varleq CN^{-D}. (2.26)

We fix now a certain index set. Let 0<q<1. Let

 \mathcal{C}_{q}:=\{i:V_{i}\in\mathcal{I}_{q}\}. (2.27)

#### 2.4.1 Classical eigenvalue locations

Given a probability measure \rho(x)\mathrm{d}x and matrix size N, we define the classical eigenvalues \gamma_{i}=\gamma_{i,N} in the following manner. If N is even then

 \gamma_{i}=\inf\left\{x:\int_{-\infty}^{x}\rho(E)\mathrm{d}E\vargeq\frac{i+1}{% N}\right\} (2.28)

and if N is odd then

 \gamma_{i}=\inf\left\{x:\int_{-\infty}^{x}\rho(E)\mathrm{d}E\vargeq\frac{i+1/2% }{N}\right\}. (2.29)

We denote the classical eigenvalue locations of the free convolution law at times t by \gamma_{i}(t) and the classical eigenvalue locations of the semicircle law by \gamma^{(\mathrm{sc})}_{i}. The above definition is slightly nonstandard, but we take it so that

 \gamma^{(\mathrm{sc})}_{\lceil N/2\rceil}=0, (2.30)

which will turn out to be convenient later.

#### 2.4.2 Rigidity estimates

We have the following rigidity result for the eigenvalues.

###### Theorem 2.7.

Fix 0<q<1 and let t be as above. For any \varepsilon>0 and D>0 we have

 \mathbb{P}\left[\sup_{i\in\mathcal{C}_{q}}|\lambda_{i}(t)-\gamma_{i}(t)|% \vargeq\frac{N^{\varepsilon}}{N}\right]\varleq N^{-D}. (2.31)

We also have

 \mathbb{P}\left[\sup_{j}|\lambda_{j}-V_{j}|\vargeq 3\sqrt{t}\right]\varleq N^{% -D}. (2.32)

From the above theorem we see that for any q_{1} with q<q_{1}<1 we have for N large enough,

 i\in\mathcal{C}_{q}\implies\gamma_{i}(t)\in\mathcal{I}_{q_{1}}. (2.33)

### 2.5 Proof strategy

In this section we give an overview of the strategy of the proof of fixed energy universality. The DBM flow starting from V is given by the SDE

where the \mathrm{d}B_{i} are standard Brownian motions.

1. Regularization. In the next step we will couple the DBM (2.34) to an auxilliary process. Before this coupling, we first allow the DBM (2.34) to run freely for an initial time interval of length t_{0}, where t_{0} satisfies the compatibility conditions g\ll t_{0}\ll G^{2}. This is needed for several reasons. Firstly, after t_{0}, we can apply the results [55] which state that rigidity holds wrt the free convolution law. Secondly, this regularizes the DBM flow in the sense that the free convolution will be regular on this scale; for example |\rho_{\mathrm{fc}}^{\prime}(E)|\varleq C/t_{0}.

2. Matching and coupling. For times t\vargeq t_{0} we couple the DBM flow to another DBM flow started from an independent GOE ensemble. That is, we define the process

 \mathrm{d}y_{i}(t)=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum_{j}\frac{% 1}{y_{i}-y_{j}}\mathrm{d}t (2.35)

where initially y_{i}(t_{0}) is distributed as a GOE ensemble independent from \{x_{i}\}_{i}. The point is that the Brownian motions in (2.34) and (2.35) are the same. This idea first appeared in [17]. Moreover, we re-scale and shift the DBM flow so that the classical eigenvalue locations match those of the semicircle law near a chosen energy E. Due to the regularity of the free convolution law, we can match up to \sqrt{Nt_{0}} eigenvalues. This matching implies that

 x_{i}(t)-y_{i}(t)\sim\frac{\log(N)}{N}, (2.36)

for eigenvalues that are near the spectral energy E. At this point we are now regarding t_{0} as a fixed a-priori scale on which the DBM flow is regular. By running the coupling for times t satisfying t_{0}\varleq t\varleq t_{0}+t_{1} with t_{1}\ll t_{0}, the DBM flow will not see the non-matching eigenvalues.

3. Discrete parabolic equation. The difference w_{i}(t):=x_{i}(t)-y_{i}(t) satisfies the parabolic equation

 \partial_{t}w_{i}=(\mathcal{L}w)_{i} (2.37)

where

 (\mathcal{L}w)_{i}:=\frac{1}{N}\sum_{j}\frac{w_{j}-w_{i}}{(x_{i}-x_{j})(y_{i}-% y_{j})}. (2.38)

As N\to\infty, a natural limit for this equation is

 \partial_{t}f(x)=\int\frac{f(y)-f(x)}{(x-y)^{2}}\rho_{\mathrm{fc},t}(y)\mathrm% {d}y\sim\int\frac{f(y)-f(x)}{(x-y)^{2}}\mathrm{d}y (2.39)

In order to justify the replacement of \rho_{\mathrm{fc},t} by a constant we will use its regularity on the scale t_{0} and a short-range approximation of the DBM flow. We omit the details in this simple sketch.

4. Homogenization theory. We may now write

 x_{i}(t_{0}+t)-y_{i}(t_{0}+t)=\sum_{j}\mathcal{U}^{\mathcal{L}}_{ij}(t_{0},t_{% 0}+t)(x_{j}(t_{0})-y_{j}(t_{0})), (2.40)

where \mathcal{U}^{\mathcal{L}} is the semigroup for the equation (2.38). We need to develop a homogenization theory in order to calculate the matrix elements \mathcal{U}^{\mathcal{L}}_{ij}. We let p_{t}(x,y) denote the fundamental solution of (2.39). Let

 f_{i}(t):=p_{t}(x_{i}(t),\gamma_{a}) (2.41)

and

 g_{i}(t):=N\mathcal{U}^{\mathcal{L}}_{i,a}(t_{0}+t,t_{0}). (2.42)

Our main calculation is

 \mathrm{d}||f-g||_{2}^{2}=-c\langle(f-g),\mathcal{L}(f-g)\rangle\mathrm{d}t+% \mathrm{d}M_{t}+\frac{o(1)}{t^{2}}\mathrm{d}t, (2.43)

where M is a martingale. Integrating this inequality in time (and dropping the time average for simplicity) we will obtain

 \langle(f-g),\mathcal{L}(f-g)\rangle\varleq\frac{o(1)}{t^{2}}+\frac{1}{t}||g(0% )-f(0)||_{2}^{2} (2.44)

The second term on the RHS is not well-defined, as f(0) is a delta function (as a distribution on \mathbb{R}) and g(0) is a discrete delta function. In order to make sense of this quantity, we introduce an additional regularization to the initial data f(0),g(0). We omit the details from this sketch but say that this roughly corresponds to convolving f(0) and g(0) with a mollifier which lives on a regularization scale s_{r} (we will only use this notation of s_{r} in this sketch and it is absent from the remainder of the paper). This mollification allows us to take ||g(0)-f(0)||_{2}^{2}=o(t^{-1}). By choosing the regularization scale s_{r}\ll t, we will also see that the regularization does not significantly affect the final value f(t) and g(t) (as the regularization/mollification scale s_{r} is shorter than the natural scale t of these functions).

We obtain

 \langle(f(t)-g(t)),\mathcal{L}(f(t)-g(t))\rangle\varleq\frac{o(1)}{t^{2}}. (2.45)

Using a discrete \dot{H}^{1/2}-\ell^{\infty} Sobolev inequality, this inequality will imply

 ||g_{i}(t)-f_{i}(t)||_{\infty}\varleq\frac{N^{-\mathfrak{b}}}{t}, (2.46)

for some positive \mathfrak{b}>0,

5. Cut-offs. The natural size of g_{i}=N\mathcal{U}^{\mathcal{L}}_{ia}(t_{0},t_{0}+t) is 1/t for indices i near a. Hence, the estimate (2.46) determines the object \mathcal{U}^{\mathcal{L}}_{ia} beyond its natural scale and we can use it to control the terms in the sum (2.40) for j near i; that is, we can control approximately o(N^{\mathfrak{b}}Nt) terms using (2.46) and (2.36).

For terms satisfying NtN^{\mathfrak{b}/2}<|i-j|<\sqrt{Nt_{0}} we have the estimate (2.36) as well as the a-priori upper bound for \mathcal{U}^{\mathcal{L}},

 N\mathcal{U}^{\mathcal{L}}_{ij}(t_{0}+t,t_{0})\varleq N^{\varepsilon}p_{t}(% \gamma_{i},\gamma_{j})\varleq N^{\varepsilon}\frac{t}{t^{2}+(\gamma_{i}-\gamma% _{j})^{2}} (2.47)

for any small \varepsilon>0. This allows us to control the contribution to the sum (2.40) for terms in this range.

Finally we have to deal with the contribution of j so that |j-i|>\sqrt{Nt_{0}}. Here, we do not have the estimate (2.36) due to lack of sufficient regularity of the initial data x_{i}(0), and the fact that the decay (2.47) is not fast enough to counteract the growth of the LHS of (2.36) as j moves further from i. Instead, we will modify the processes (2.34) and (2.35) and replace them by certain short-range approximations. For the purposes of this sketch we will not define the approximations precisely. We will just say that the local law and rigidity estimates allow us to replace the long-range contribution in (2.34) and (2.35) of terms with |i-j|>\ell large with a deterministic drift term. Here \ell is an additional scale chosen larger than Nt. This modifies the operator \mathcal{L} to only allow jumping between sites |i-j|<\ell. The behavior of the kernal \mathcal{U}^{\mathcal{L}} is then modified to decay exponentially for |i-j|>\sqrt{Nt}\ell (i.e., simple random walk behavior in 1 dimension with step-size \ell). This latter property enables the cut-off of the non-matching terms where the estimate (2.36) fails.

6. Mesoscopic linear statistics. The homogenization theory proves that there is a smooth function \zeta_{t}:\mathbb{R}\to\mathbb{R} so that (up to errors)

 x_{i}(t_{0}+t)-y_{i}(t_{0}+t)=\sum_{|i-j|\varleq Nt}\zeta_{t}(\gamma_{i}-% \gamma_{j})(x_{j}(t_{0})-y_{j}(t_{0}))=:\zeta_{x}-\zeta_{y}. (2.48)

Roughly, \zeta_{t}(\gamma_{i}-\gamma_{j})\sim p_{t}(\gamma_{i},\gamma_{j}) where p_{t} is the fundamental solution introduced above (to the PDE (2.39)). This reduces the microscopic problem to a simpler mesoscopic one. For this mesoscopic observable, we calculate the characteristic function and prove that

 \left|\psi(\lambda)\right|:=\left|\mathbb{E}[\mathrm{e}^{\mathrm{i}\lambda% \zeta_{x}}]\right|\varleq\mathrm{e}^{-\lambda^{2}c_{x}\log(N)}+N^{-\varepsilon} (2.49)

for some \varepsilon>0 and a constant c_{x}>0.

We remark that we will use a slightly different convention for \zeta_{t} in the full proof than in the simple sketch given here. The precise definition of \zeta_{t} is given Section 3. To differentiate the two conventions we use \zeta(x,t) instead of \zeta_{t}(x) in the rest of the paper.

7. Fourier cut-off. We now proceed similarly to [17]. It suffices to consider for smooth test functions Q sums of terms of the form,

 \mathbb{E}[Q(N(x_{i}-E),N(x_{j}-x_{i}))]. (2.50)

The homogenization theory shows that

 \mathbb{E}[Q(N(x_{i}-E),N(x_{j}-x_{i}))]=\mathbb{E}[Q(N(y_{i}-E+(\zeta_{x}-% \zeta_{y})),N(y_{j}-y_{i}))]+o(1). (2.51)

By Fourier duality we have

 \mathbb{E}[Q(N(y_{i}-E+(\zeta_{x}-\zeta_{y})),N(y_{j}-y_{i}))]=\int\mathbb{E}[% \hat{Q}(\lambda,N(y_{i}-y_{j}))\mathrm{e}^{\mathrm{i}\lambda N(y_{i}-E+\zeta_{% x}-\zeta_{y})}]\psi(\lambda)\mathrm{d}\lambda. (2.52)

Here \hat{Q} denotes the Fourier transform of Q in the first variable. By (2.49) we can cut off the Fourier support of \hat{Q} in the range |\lambda|\vargeq\delta for any small fixed \delta>0. Therefore it suffices to consider observables Q with Fourier support contained in |\lambda|\varleq\delta.

8. Reverse heat-flow. Running the same argument with a third ensemble z distributed as the GOE shows that

 \mathbb{E}[Q(N(z_{i}-E),N(z_{j}-z_{i}))]=\mathbb{E}[Q(N(y_{i}-E+(\zeta_{z}-% \zeta_{y})),N(y_{j}-y_{i}))]+o(1). (2.53)

As in [17] we see that from (2.51) and (2.53) that fixed energy universality will follow if we can prove that the function

 F(a):=\mathbb{E}[Q(N(y_{i}-E+(a-\zeta_{y})),N(y_{j}-y_{i}))] (2.54)

is approximately constant, for Q a function of small Fourier support. The argument to prove this is the same as in [17]. What is new is that we have analyzed the mesoscopic statistic \zeta_{x} and used it to complete the Fourier cut-off in the previous step. In [17] a Fourier cut-off was also used, but only for \delta a large constant; here, \delta is allowed to be any small constant. In [17] this caused some restriction in the following argument on how small t can be in proving fixed energy universality. Here this restriction is removed due to the Fourier cut-off \delta.

We would like to prove that F(a) is constant. Define F_{h}(a):=F(a+h)-F(a). By translation invariance of the local GOE statistics we know that \mathbb{E}[F_{h}(\zeta_{z})]=o(1). We will prove that \zeta_{z} is close to a Gaussian with variance c_{z}\log(N). In order to conclude that F_{h} is small we run the reverse heat flow argument of [17]. We see that

 \left|\hat{F}_{h}(\lambda)\mathrm{e}^{-c_{z}\log(N)\lambda^{2}}\right|\varleq N% ^{-\varepsilon} (2.55)

for some \varepsilon>0, independent of the \delta chosen above. By the Fourier support restriction on Q we see that |\hat{F}_{h}(\lambda)|\varleq N^{\delta^{2}c_{z}}N^{-\varepsilon}. Hence for \delta small enough we get that \hat{F}_{h}=o(1) and we conclude that F_{h} is small. This proves fixed energy universality.

### 2.6 Notation

We will use the following notion of overwhelming probability.

###### Definition 2.8.

We say that an event \mathcal{F} holds with overwhelming probability if for any D>0 we have \mathbb{P}[\mathcal{F}^{c}]\varleq N^{-D} for large enough N. If we have a family of events \{\mathcal{F}(u)\}_{u} then we will say that \{\mathcal{F}(u)\}_{u} holds with overwhelming probability if \sup_{u}\mathbb{P}[\mathcal{F}^{c}(u)]\varleq N^{-D} for large enough N.

For two positive N-dependent quantities a_{N} and b_{N} we say that a_{N}\asymp b_{N} if there are constants c and C s.t. ca_{N}\varleq b_{N}\varleq Ca_{N}.

In our work we use C to denote a positive constant that can change from line to line. The constant C will typically only depend on the constants appearing in the assumptions on V.

For A,B\in\mathbb{R} we denote

 [[A,B]]:=[A,B]\cap\mathbb{Z}. (2.56)

## 3 Homogenization

In this section we prove a homogenization result for DBM. This reduces the problem of fixed energy universality of the model H_{t} to a problem involving mesoscopic statistics. Given a real symmetric matrix M with eigenvalues \lambda_{1}(M)\varleq\cdots\varleq\lambda_{N}(M), we define Dyson Brownian motion with \beta=1 and initial data M to be the process satisfying

At each fixed time t\vargeq 0, the particles \{\lambda_{i}(t)\}_{i} are equal in distribution to the eigenvalues of the matrix M+\sqrt{t}W, where W is a GOE matrix independent of M. It is well-known that there is a unique strong solution to the above system of SDEs and the sample paths are continuous a.s. Recall that we want to study the eigenvalues of the matrix H_{t}:=V+\sqrt{t}W. We will do this by studying the DBM flow for t in the regime

 t_{0}\varleq t\varleq t_{0}+t_{1}. (3.2)

Here t_{0} and t_{1} are times defined by t_{i}=N^{\omega_{i}}/N and \omega_{1}<\omega_{0}. The time t_{0} satisfies gN^{\sigma}\varleq t_{0}\varleq N^{-\sigma}G^{2}.

Our study requires the choice of an index i_{0}\in\mathcal{C}_{q}, with \mathcal{C}_{q} defined in (2.27). We will compare eigenvalues near i_{0} to the GOE. At time t_{0} the eigenvalue density of H_{t_{0}} is given by the free convolution law \rho_{\mathrm{fc},t_{0}} as defined in Section 2.1. In this section we are going to assume that

In applications of the homogenization theorem this will be implemented by a re-scaling and shift of V and a re-scaling of time. For times t\vargeq t_{0} we define x_{i}(t) to be the solution of

 \mathrm{d}x_{i}(t):=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum_{j\neq i% }\frac{1}{x_{i}(t)-x_{j}(t)}\mathrm{d}t (3.4)

with initial data x_{i}(t_{0})=\lambda_{i}(H_{t_{0}}).

We now introduce the coupled GOE process. For times t\vargeq t_{0} define y_{i}(t) as the solution to

 \mathrm{d}y_{i}(t)=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum_{j\neq i}% \frac{1}{y_{i}(t)-y_{j}(t)}\mathrm{d}t (3.5)

where the initial data y_{i}(t_{0}) are the eigenvalues of a GOE matrix independent of \{x_{i}(t)\}_{i}. Above, the Brownian motions are the same as those appearing in (3.4). At times t\vargeq t_{0} the particles \{y_{i}(t)\}_{i} are distributed as

 \{y_{i}(t)\}_{i}\overset{d}{=}\left\{\lambda_{i}\left(\sqrt{1+(t-t_{0})}W^{% \prime}\right)\right\}_{i}. (3.6)

At times t\vargeq t_{0} the y_{i}(t) satisfy a rigidity estimate with respect to the classical eigenvalue locations \sqrt{1+(t-t_{0})}\gamma^{(\mathrm{sc})}_{i} where \gamma^{(\mathrm{sc})}_{i} denote the classical eigenvalue locations of the semicircle law \rho_{\mathrm{sc}}. For our purposes we adopt the convention N/2:=\lceil N/2\rceil as well as \gamma^{(\mathrm{sc})}_{N/2}=0.

In order to state the following theorem we introduce a function \zeta(x,t):\mathbb{R}^{2}\to\mathbb{R}. It is defined in terms of a fundamental solution p_{t}(x,y) to a specific non-local integral equation that we will compare our process to. This non-local integral equation is (3.88) below; its definition requires a cut-off which will be introduced over the next part of the proof, so we defer the definition for now. For the time being it will suffice to just assert the existence of this function \zeta(x,t) and summarize some of its properties in Proposition 3.2 below. In terms of p_{t}(x,y) we have,

 \zeta(x,t)=\frac{1}{N}p_{t}(0,x/\rho_{\mathrm{sc}}(0)) (3.7)

The following theorem is the main result of this section.

###### Theorem 3.1.

Fix t_{0}=N^{\omega_{0}}/N satisfying gN^{\sigma}\varleq t_{0}\varleq N^{-\sigma}G^{2} for \sigma>0. Let t_{1}=N^{\omega_{1}}/N with 0<\omega_{1}<\omega_{0}/2. Let

 t_{2}=\max\{N^{-\omega_{1}/15}t_{1},N^{-4(\omega_{0}/2-\omega_{1})/3}t_{1}\}. (3.8)

Let 0<\varepsilon_{b}<\min\{(\omega_{0}/2-\omega_{1})/3,\omega_{1}/60\}. Let i_{0}\in\mathcal{C}_{q}. Assume that

With overwhelming probability we have the following estimates. For every |u|\varleq t_{2} and |i|\varleq Nt_{1}N^{\varepsilon_{b}} we have

 \displaystyle\left(x_{i_{0}+i}(t_{0}+t_{1}+u)-\gamma_{i_{0}}(t_{0}+t_{1}+u)% \right)-y_{N/2+i}(t_{0}+t_{1}+u) \displaystyle= \displaystyle\sum_{|j|\varleq N^{\omega_{1}+(\omega_{0}/2-\omega_{1})/3}}\zeta% \left(\frac{i-j}{N},t_{1}\right)\left[x_{i_{0}+j}(t_{0})-y_{N/2+j}(t_{0})% \right]+\frac{N^{\varepsilon}}{N}\mathcal{O}\Bigg{(}\frac{N^{\omega_{1}/3}}{N^% {\omega_{0}/6}}+\frac{1}{N^{\omega_{1}/60}}\Bigg{)} (3.10)

Theorem 3.1 will be a consequence of Theorem 3.7 below. For the mesoscopic statistic \zeta(x,t_{1}) we have the following properties.

###### Proposition 3.2.

The function \zeta(x,t) satisfies for 0\varleq t\varleq 1 and x\in\mathbb{R} the following. We have,

and

### 3.1 Re-indexing

In this subsection we are going to make some assumptions which will greatly simplify notation. In Appendix C we present an argument which reduces the general case to these assumptions.

Let i_{0} be as in Theorem 3.1. We assume that N is odd and that i_{0}=(N+1)/2. Note that with our convention, \gamma^{(\mathrm{sc})}_{i_{0}}=0.

Presently the eigenvalues are labelled by the integers [[1,N]]. We re-label the eigenvalues so that they are indexed by [[-(N-1)/2,(N-1)/2]]. The eigenvalues are then x_{-(N-1)/2}\varleq x_{-(N-3)/2}\varleq\cdots\varleq x_{(N-1)/2} and i_{0}=0. We furthermore have that \gamma_{0}(t_{0})=0.

We adopt these assumptions for the remainder of Section 3, apart from Section 3.8 which is where we prove Theorem 3.1 and must therefore unravel the re-labelling. We also adopt this convention for Section 4. It will not be used in the other sections.

### 3.2 Interpolation

In the work [17], the parabolic equation satisfied by the differences u_{i}:=x_{i}-y_{i} was directly considered. The jump rates of the generator of this equation are hard to control as they involve both of the differences (x_{i}-x_{j}) and (y_{i}-y_{j}). In this paper we define a continuous interpolation which allows us to consider a family of parabolic equations whose generators are easier to control.

We now introduce this interpolation. For 0\varleq\alpha\varleq 1, we define z_{i}(t,\alpha) as the solution to

 \mathrm{d}z_{i}(t,\alpha)=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\left(\frac{1}{N}% \sum_{j}\frac{1}{z_{i}(t,\alpha)-z_{j}(t,\alpha)}\right)\mathrm{d}t,\qquad z_{% i}(0,\alpha)=\alpha x_{i}(t_{0})+(1-\alpha)y_{i}(t_{0}). (3.13)

Note that z_{i}(t,0)=y_{i}(t_{0}+t) and z_{i}(t,1)=x_{i}(t_{0}+t). Note that we have effectively introduced a time shift which sets t_{0}=0. In the remainder of Section 3 we will refer to \{z_{i}(t,\alpha)\}_{i} as “particles,” instead of using the terminology of eigenvalues. Below we will introduce some other processes with similar notation that we will also refer to as particles.

We will soon see that like the x_{i} and y_{i}, the z_{i}(t,\alpha) satisfy a rigidity estimate. However, we have some freedom in choosing the measure with which to construct the classical particle (eigenvalue) locations (note that here we are still referring to quantiles of a measure with which the empirical density of the z_{i}(t,\alpha) but in accordance with calling the z_{i}(t,\alpha) particles we will call them classical particle locations). One choice is the free convolution of the empirical measure of the initial data z_{i}(0,\alpha) with the semicircle law. However, this law is somewhat singular for short times t, and does not reflect the fact that at time t_{0}, the particles x_{i}(t_{0}) satisfy a rigidity estimate wrt \rho_{\mathrm{fc}}(t_{0}) which has some regularity properties (e.g., |\rho_{\mathrm{fc}}^{\prime}(t_{0})|\varleq C/t_{0}).

To compensate for this, we construct a measure \nu(\mathrm{d}x,\alpha) that has a density near 0 that is at least as smooth as \rho_{\mathrm{fc}}(t_{0}). The construction is described in detail in Appendix A, and for now we just sketch its construction. This measure constructed is random but has good properties with overwhelming probability.

We need \nu(\mathrm{d}x,\alpha) to satisfy two properties which motivate its construction. Firstly, we would like it to have a smooth density near 0 which is at least as regular as \rho_{\mathrm{fc}}(t_{0}). Secondly, we need the initial data z_{i}(0,\alpha) to be approximated by \nu(\mathrm{d}x,\alpha) down to the optimal scale \eta\gtrsim 1/N, so that at later times t, the particles z_{i}(t,\alpha) follow the free convolution of \nu(\mathrm{d}x,\alpha) with the semicircle distribution.

We now sketch the construction of the measure \nu(\mathrm{d}x,\alpha); complete details are given in Appendix A. The construction requires the choice of a parameter 0<q^{*}<1 which we now fix. This q^{*} is the same as that which appears in Appendix A. In the interval [-q^{*}G,q^{*}G], one can construct, using the inverse function theorem, a density whose quantiles (in this case defined by starting the integration of the density from 0) equal \alpha\gamma_{i}(t_{0})+(1-\alpha)\gamma^{(\mathrm{sc})}_{i}. This density has as good regularity properties as \rho_{\mathrm{fc}}(t_{0}). Since z_{i}(0,\alpha) satisfy a rigidity estimate in the interval [-q^{*}G,q^{*}G], this density gives the required approximation for z_{i} in this interval. To approximate the z_{i} outside the interval [-q^{*}G,q^{*}G] we can take \nu(\mathrm{d}x,\alpha) to consist of a Dirac delta mass at each z_{i}(t,\alpha) such that |z_{i}(t,\alpha)|>q^{*}G. Since the delta functions are outside of the interval [-q^{*}G,q^{*}G] we do not affect the regularity inside this interval. Clearly \nu(\mathrm{d}x,\alpha) gives a good approximation to z_{i}(t,\alpha).

We now record formally some of the needed properties of the measure \nu(\mathrm{d}x,\alpha). Again, we mention that the explicit construction appears in Appendix A. One of the key properties will be that, although the measure \nu(\mathrm{d}x,\alpha) and its free convolution are random, for \alpha=0,1 the quantitative properties inside |x|\varleq q^{*}G coincide with \rho_{\mathrm{fc},t} and \rho_{\mathrm{sc}} and are deterministic. In particular, certain quantiles of the measure at later times are deterministic up to an error term. This is summarized in Lemma 3.4 below.

Let k_{0} be the largest index so that

Note that k_{0}\asymp NG. The measure \nu(\mathrm{d}x,\alpha) has a nonvanishing density on the interval

 \mathcal{G}_{\alpha}:=[\alpha\gamma_{-k_{0}}(t_{0})+(1-\alpha)\gamma^{(\mathrm% {sc})}_{-k_{0}}(t_{0}),\alpha\gamma_{k_{0}}(t_{0})+(1-\alpha)\gamma^{(\mathrm{% sc})}_{k_{0}}] (3.15)

but has a singular part which may overlap with an o(G) portion of \mathcal{G}_{\alpha} at its boundary; for any 0<q<1, \nu(\mathrm{d}x,\alpha) is purely a.c. on q\mathcal{G}_{\alpha}.

We now define \rho(E,t,\alpha) to be the free convolution of \rho(E,0,\alpha)\mathrm{d}E:=\nu(\mathrm{d}E,\alpha) and the semicircle distribution with Stieltjes transform m(z,t,\alpha). We have abused notation here slightly as at time t=0, the measure \nu(\mathrm{d}x,\alpha) is the sum of an absolutely continuous part and a sum of delta functions and it is only for times t>0 that it has a true density. However, this will not affect anything as whenever we write \rho(E,0,\alpha) we only be referring to E near 0 where the measure \nu(\mathrm{d}x,\alpha) is purely a.c.

The following holds for the free convolutions. We defer the proof to Appendix A.

###### Lemma 3.3.

Let \delta>0. All of the following holds for |E|\varleq N^{-\delta}t_{0} and t\varleq N^{-\delta}t_{0}, and N^{\delta}/N\varleq\eta\varleq 10, and with overwhelming probability. For the Stieltjes transform we have

 |m(t,E,\alpha)-m(t,0,\alpha)-(m(t,E,0)-m(t,0,0))|\varleq C\log(N)\left(\frac{|% E|}{t_{0}}+\frac{t}{t_{0}}\right) (3.16)

We have

 |\partial_{z}m(z,t,\alpha)|\varleq\frac{C}{t_{0}}. (3.17)

For the free convolution laws we have

and

 \left|\rho(E,t,\alpha)-\rho_{\mathrm{sc}}(0)\right|\varleq C\log(N)\frac{|E|+t% }{t_{0}}. (3.19)

Moreover, for 0<q<1 and E\in q\mathcal{G}_{\alpha}, N^{\delta}/N\varleq\eta\varleq 10,

 |m(z,t,\alpha)|\varleq C\log(N),\qquad c\varleq\mathrm{Im}\mbox{ }[m(t,z,% \alpha)]\varleq C. (3.20)

The classical particle locations of the measure \rho(E,t,\alpha) are denoted by \gamma_{i}(t,\alpha) and are defined by

 \frac{1}{2}+\frac{i}{N}=\int_{-\infty}^{\gamma_{i}(t,\alpha)}\rho(E,t,\alpha)% \mathrm{d}E. (3.21)

We have that \gamma_{0}(0,\alpha)=0 by the definition of \rho(E,0,\alpha). We will also need to relate \gamma_{i}(t,0) and \gamma_{i}(t,1) back to \gamma^{(\mathrm{sc})}_{i} and \gamma_{i}(t), respectively.

###### Lemma 3.4.

We have for any \varepsilon>0 and \omega_{1}<\omega_{0}/2 with overwhelming probability,

 \sup_{0\varleq t\varleq 10t_{1}}|\gamma_{0}(t,1)-\gamma_{0}(t_{0}+t)|\varleq% \frac{N^{\varepsilon}}{N}\frac{N^{\omega_{1}}}{N^{\omega_{0}/2}} (3.22)

and

 \sup_{0\varleq t\varleq 10t_{1}}|\gamma_{0}(t,0)-0|\varleq\frac{N^{\varepsilon% }}{N}\frac{N^{\omega_{1}}}{N^{\omega_{0}/2}}. (3.23)

For |j|,|k|\varleq N^{\omega_{0}/2} we have with overwhelming probability,

 \gamma_{k}(t,\alpha)-\gamma_{j}(t,\alpha)=\frac{1}{N}\frac{k-j}{\rho_{\mathrm{% sc}}(0)}+\mathcal{O}\left(\frac{1}{N}\right), (3.24)

for any t\varleq 10t_{1}, with \omega_{1}\varleq\omega_{0}/2 and 0\varleq\alpha\varleq 1.

We again defer the proof to Appendix A.

We define the empirical Stieltjes transforms by

 m_{N}(z,t,\alpha)=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{z_{i}(t,\alpha)-z}. (3.25)

The measures \rho(E,t,\alpha) are \alpha dependent. We are eventually going to introduce some short-range and long-range cut-offs, and differentiate certain objects in \alpha. As the cut-offs are inherently discrete (they involve the particle indices), we want to choose them independent of \alpha to interact nicely with the differentiation. This requires the introduction of the following index sets. Let k_{1} be the largest index so that

 \bigcup_{0\varleq\alpha\varleq 1}[\alpha\gamma_{-k_{1}}(t_{0})+(1-\alpha)% \gamma^{(\mathrm{sc})}_{-k_{1}}(t_{0}),\alpha\gamma_{k_{1}}(t_{0})+(1-\alpha)% \gamma^{(\mathrm{sc})}_{k_{1}}]\subseteq\bigcap_{0\varleq\alpha\varleq 1}% \mathcal{G}_{\alpha}\cap\{-\mathcal{G}_{\alpha}\} (3.26)

where -\mathcal{F}:=\{x:-x\in\mathcal{F}\}.

Finally, for any 0<q<1, we define

 \hat{\mathcal{C}}_{q}=\{j:|j|\varleq qk_{1}\}. (3.27)

The cardinality \hat{\mathcal{C}}_{q} satisfies |\hat{\mathcal{C}}_{q}|\asymp qGN. This definition is just so that particles z_{k}(t,\alpha), for k\in\hat{\mathcal{C}}_{q} have nice qualitative properties uniformly in \alpha (the constants degenerate as q\to 1). For example, the optimal rigidity estimate holds for any k\in\hat{\mathcal{C}}_{q}. Moreover, the classical particle locations \gamma_{k}(t,\alpha) for k\in\hat{\mathcal{C}}_{q} will all be contained in a symmetric interval [-q^{\prime}G,q^{\prime}G] for some q^{\prime} on which all the densities \rho(E,t,\alpha) have good properties.

We will also use tacitly that for any q we have for j,k\in\hat{\mathcal{C}}_{q} that

 c\frac{|j-k|}{N}\varleq|\gamma_{j}(t,\alpha)-\gamma_{k}(t,\alpha)|\varleq C% \frac{|j-k|}{N}. (3.28)

We have the following rigidity and local law estimates.

###### Lemma 3.5.

Let \varepsilon>0, \delta>0, \delta_{1}>0, D>0 and 0<q<1. We have

 \mathbb{P}\left[\sup_{0\varleq t\varleq N^{-\delta_{1}}t_{0}}\sup_{i\in\hat{% \mathcal{C}}_{q}}\sup_{0\varleq\alpha\varleq 1}|z_{i}(t,\alpha)-\gamma_{i}(t,% \alpha)|\vargeq\frac{N^{\varepsilon}}{N}\right]\varleq N^{-D}. (3.29)

We have also,

 \mathbb{P}\left[\sup_{N^{\delta}/N\varleq\eta\varleq 10}\sup_{0\varleq t% \varleq N^{-\delta_{1}}t_{0}}\sup_{0\varleq\alpha\varleq 1}\sup_{E\in q% \mathcal{G}_{\alpha}}|m_{N}(z,t,\alpha)-m(z,t,\alpha)|\vargeq\frac{N^{% \varepsilon}}{N\eta}\right]\varleq N^{-D}. (3.30)

We defer the proof to Appendix A.

#### 3.2.1 Reformulation of homogenization

We have proven already a few estimates about the processes \{z_{i}(t,\alpha)\}_{i} introduced above. In order to clarify what properties of the processes we are using we are going to reformulate the relevent estimates as hypotheses and use them to prove a theorem. This theorem will then be used to prove Theorem 3.1. Lemmas 3.3 and 3.5 imply that the hypotheses 1-4 are satisfied by the processes \{z_{i}(t,\alpha)\}_{i} considered above.

The point of this is to give a black box result which can be applied to other DBM processes. Essentially the two key inputs are the rigidity estimates and the regularity of the free convolution measure that the particles are being compared to. Here, we have proven the rigidity estimates using the fact that the process comes from a matrix model. In other settings, e.g., non-classical \beta, such methods may not be available and the rigidity could be proven by other means. The regularity of the measure \rho_{\mathrm{fc},t_{0}} describing the empirical eigenvalue distribution of x_{i}(t_{0}) comes from the fact that we were able to run the DBM for a regularization period before running the coupling; however, the initial data under consideration in different contexts may obey this regularity condition due to other reasons. In such cases, the following reformulation will apply.

Our starting point is that there are two processes z_{i}(t,0) and z_{i}(t,1) which satisfy (3.13) (for \alpha=0,1, of course) with initial data which we denote by z_{i}(0,0) and z_{i}(0,1). Given these two processes, we then construct the interpolating processes z_{i}(t,\alpha) by (3.13), except that the initial data is z_{i}(0,\alpha)=\alpha z_{i}(0,1)+(1-\alpha)z_{i}(0,0). We assume the following on the processes z_{i}(t,\alpha).

1. [label=()]

2. z_{i}(0,0) is a GOE ensemble.

3. There is a law \rho(E) with Stieltjes transform m(z) and parameters t_{0}=N^{\omega_{0}}/N and G with t_{0}\varleq N^{-\sigma}G^{2} so that the following hold.

The classical particle locations of \rho(E) satisfy \gamma_{0}=0 and \rho(0)=\rho_{\mathrm{sc}}(0).

4. We need also measures that give the empirical particle density of the z_{i}(t,\alpha). For this, we proceed the same as in Section 3.2, with \rho(E) from the previous assumption taking the place of \rho_{\mathrm{fc},t_{0}}. That is, we fix a 0<q^{*}<1, and first construct \nu(dE,\alpha) as a density on [-q^{*}G,q^{*}G], add to it some delta functions and then take the free convolution. We denote the resulting measure by \rho(E,t,\alpha)dE. We then use much of the same notation as introduced in Section 3.2; we have the classical particle locations \gamma_{i}(t,\alpha), Stieltjes transform m(z,t,\alpha) and index sets \hat{\mathcal{C}}_{q} and energy windows \mathcal{G}_{\alpha}.

The assumption is then the following. We assume the following rigidity and local laws,

 \mathbb{P}\left[\sup_{0\varleq t\varleq N^{-\delta_{1}}t_{0}}\sup_{i\in\hat{% \mathcal{C}}_{q}}\sup_{0\varleq\alpha\varleq 1}|z_{i}(t,\alpha)-\gamma_{i}(t,% \alpha)|\vargeq\frac{N^{\varepsilon}}{N}\right]\varleq CN^{-D} (3.32)

and

 \mathbb{P}\left[\sup_{0\varleq t\varleq N^{-\delta_{1}}t_{0}}\sup_{N^{\delta-1% }\varleq\eta\varleq 10}\sup_{0\varleq\alpha\varleq 1}\sup_{E\in q\mathcal{G}_{% \alpha}}|m_{N}(z,t,\alpha)-m(z,t,\alpha)|\vargeq\frac{N^{\varepsilon}}{N\eta}% \right]\varleq CN^{-D} (3.33)

for any \delta,\varepsilon,\delta_{1},D>0 and 0<q<1.

5. There is a C>0 so that for any D>0,

 \mathbb{P}\left[\sup_{0\varleq t\varleq t_{0}}\sup_{i}|z_{i}(t,1)|\vargeq N^{C% }\right]\varleq N^{-D}. (3.34)

for large enough N.

Since the only properties of \rho_{\mathrm{fc},t_{0}} used in the proofs of Lemma 3.3 and (3.24) are those assumed in 2, the estimates hold for new measures \rho(E,t,\alpha)dE and their quantiles as introduced in 3. We record this in the following lemma.

###### Lemma 3.6.

Under the above set-up and assuming 1-4, the estimates of Lemma 3.3 and the estimate (3.24) hold for the measures \rho(E,t,\alpha) and the classical particle locations \gamma_{i}(t,\alpha).

Under the above assumptions we will prove the following, from which Theorem 3.1 will be deduced.

###### Theorem 3.7.

Let z_{i}(t,0) and z_{i}(t,1) be defined as at the start of Section 3.2.1. Suppose that assumptions 1-4 hold. Let 0<\varepsilon_{b}<\varepsilon_{a}. Let t_{2}:=t_{1}N^{-\varepsilon_{2}} with \omega_{1}-\varepsilon_{1}>0. Let \varepsilon>0. There is a parameter \ell=N^{\omega_{\ell}} so that the following holds (see (3.41) below for its definition). There is an event with overwhelming probability on which the following holds. For any |i|\varleq Nt_{1}N^{\varepsilon_{b}} and |u|\varleq t_{2} we have

 \displaystyle z_{i}(t_{1}+u,1)-z_{i}(t_{1}+u,0)=\left(\gamma_{0}(t_{1}+u,1)-% \gamma_{0}(t_{1}+u,0)\right) \displaystyle+ \displaystyle\sum_{|j|\varleq Nt_{1}N^{\varepsilon_{a}}}\zeta\left(\frac{i-j}{% N},t_{1}\right)(z_{j}(0,1)-z_{j}(0,0)) \displaystyle+ \displaystyle\frac{N^{\varepsilon}}{N}\mathcal{O}\left(N^{\omega_{1}}\left(% \frac{N^{\omega_{A}}}{N^{\omega_{0}}}+\frac{1}{N^{\omega_{\ell}}}+\frac{1}{% \sqrt{NG}}\right)+\frac{1}{N^{\varepsilon_{a}}}+N^{\varepsilon_{2}+\varepsilon% _{a}}\left(\frac{(Nt_{1})^{2}}{\ell^{2}}+\frac{1}{(Nt_{1})^{1/10}}\right)+N^{% \varepsilon_{a}-\varepsilon_{2}/2}\right) (3.35)

Remark. In the set-up considered here, there is no assumption about whether or not the initial data z_{i}(0,1) satisfy the local law with respect to \rho(E). In the case that this is true, the analog of (3.22) holds. That is, consider the free convolution of \rho(E) with the semicircle distribution at time t and its quantiles \gamma_{i}(t). Note that this is in general different than \rho(E,t,1) and \gamma_{i}(t,1). Under the assumption that the local law holds for z_{i}(0,1) with respect to \rho(E) and the regularity assumption 2 we see that the estimate (3.22) for the new quantities \gamma_{0}(t,1) and \gamma_{0}(t). On the other hand, the estimate (3.23) holds for the new quantity \gamma_{0}(t,0) as z_{i}(0,0) obey a local law with respect to the semicircle distribution. Hence, the classical particle locations appearing above can be replaced by deterministic counterparts which do not depend on the realization of the initial data z_{i}(0,1), under the assumption of a local law for the initial data z_{i}(0,1). ∎

### 3.3 Short-range DBM

The centered \tilde{z}_{i}’s are given by

 \tilde{z}_{i}(t,\alpha):=z_{i}(t,\alpha)-\gamma_{0}(t,\alpha). (3.36)

We also define the classical locations of the centered \tilde{z}_{i}(t,\alpha) by

 \tilde{\gamma}_{i}(t,\alpha)=\gamma_{i}(t,\alpha)-\gamma_{0}(t,\alpha). (3.37)

Note that \gamma_{i}(t,\alpha) satisfies (see [57])

 \partial_{t}\gamma_{i}(t,\alpha)=-\mathrm{Re}[m(\gamma_{i}(t,\alpha),t,\alpha)] (3.38)

and so for i\in\hat{\mathcal{C}}_{q} we have

 |\partial_{t}\gamma_{i}(t,\alpha)|\varleq C\log(N) (3.39)

by Lemma 3.6. The \tilde{z}_{i}(t,\alpha) satisfy the equations

 \mathrm{d}\tilde{z}_{i}(t,\alpha)=\frac{\mathrm{d}B_{i}}{\sqrt{N}}+\left(\frac% {1}{N}\sum_{j}\frac{1}{\tilde{z}_{i}(t,\alpha)-\tilde{z}_{j}(t,\alpha)}+% \mathrm{Re}[m(\gamma_{0}(t,\alpha),t,\alpha)]\right)\mathrm{d}t (3.40)

We introduce the following cut-off dynamics for the \{z_{i}\}_{i}. Its definition will use the parameters

Before defining the short-range approximation we outline the role of each of these parameters in the definition. The parameter \ell=N^{\omega_{\ell}} is the most fundamental. It is the “range” of the short-range approximation. In our short-range dynamics we allow particles i and j to interact iff |i-j|\varleq\ell. In order for this approximation to be effective we need \ell\gg Nt_{1}; that is \ell must exceed the “range” of DBM which is t_{1}.

For particles i near 0 we can use the rigidity estimates to replace the long-range part of the dynamics (i.e., the force coming from particles j s.t. |i-j|>\ell) by a deterministic quantity. Since the free convolution law is regular the dependence of this deterministic quantity on the particle index i is smooth; we can therefore replace it by something independent of the particle index i if |i|\varleq N^{\omega_{A}}, as long as we choose \omega_{A} to be smaller than the regularity scale of the free convolution law which is governed by t_{0}.

Finally, since the rigidity estimates only hold for particles near 0 we need to make a different cut-off for particles away from 0; this is the role of q_{*}. We will not make a short range cut-off for particles i\notin\hat{\mathcal{C}}_{q_{*}}.

We now turn to the definition of the short-range approximation. We first introduce some notation. Let \omega_{\ell}>0 be as above. For each 0<q<1, define the short range index set \mathcal{A}_{q} by

 \displaystyle\mathcal{A}_{q}:=\{(i,j):|i-j|\varleq N^{\omega_{\ell}}\}\cup\{(i% ,j):ij>0,i\notin\hat{\mathcal{C}}_{q},j\notin\hat{\mathcal{C}}_{q}\}. (3.42)

We introduce the following notation. Let

For a fixed i let j_{\varleq,i} be the smallest index s.t. (i,j_{\varleq,i})\in\mathcal{A}_{q_{*}} and j_{\vargeq,i} be the largest index s.t. (i,j_{\vargeq,i})\in\mathcal{A}_{q_{*}}. Then define the interval

 \mathcal{I}_{i}(\alpha,t)=[\gamma_{j_{\varleq,i}}(\alpha,t),\gamma_{j_{\vargeq% ,i}}(\alpha,t)]. (3.44)

The interval \mathcal{I}_{i}(\alpha,t) corresponds to the classical spatial locations of the particles j that are allowed to interact with particle i.

Let \omega_{A}>0 be as above. We define the short-range approximation \hat{z}_{i}(t,\alpha) as the solution to the following system of SDEs. For |i|\varleq N^{\omega_{A}} let

 \displaystyle\mathrm{d}\hat{z}_{i}(t,\alpha) \displaystyle=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum^{\mathcal{A}_{% q_{*}},(i)}_{j}\frac{1}{\hat{z}_{i}(t,\alpha)-\hat{z}_{j}(t,\alpha)}\mathrm{d}t (3.45)

and for |i|>N^{\omega_{A}} let

 \displaystyle\mathrm{d}\hat{z}_{i}(t,\alpha) \displaystyle=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}\sum^{\mathcal{A}_{% q_{*}},(i)}_{j}\frac{1}{\hat{z}_{i}(t,\alpha)-\hat{z}_{j}(t,\alpha)}\mathrm{d}% t+\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}^{c},(i)}_{j}\frac{1}{\tilde{z}_{i}(t,% \alpha)-\tilde{z}_{j}(t,\alpha)}\mathrm{d}t \displaystyle+\mathrm{Re}[m(t,\gamma_{0}(\alpha,t),\alpha)]\mathrm{d}t\mathrm{% d}t. (3.46)

The initial condition is \hat{z}_{i}(0,\alpha)=\tilde{z}_{i}(0,\alpha). Like the \tilde{z}_{i}(t,\alpha), the \hat{z}_{i}(t,\alpha) retain the ordering \hat{z}_{i}(t,\alpha)<\hat{z}_{i+1}(t,\alpha) for all positive times. The above parameters are chosen so that

 0<\omega_{1}<\omega_{\ell}<\omega_{A}<\omega_{0}/2. (3.47)

The following lemma shows that the \hat{z}_{i}’s are a good approximation for the z_{i}’s - that is \hat{z}_{i}=\tilde{z}_{i}+o(1/N) with overwhelming probability.

###### Lemma 3.8.

Let \hat{z}_{i}(t,\alpha) be defined as in (3.45)-(3.3) and \tilde{z}_{i}(t,\alpha) be defined as in (3.40). Let \varepsilon>0 and D>0. Then we have

 \mathbb{P}\left[\sup_{0\varleq t\varleq t_{1}}\sup_{i}\sup_{0\varleq\alpha% \varleq 1}|\hat{z}_{i}(t,\alpha)-\tilde{z}_{i}(t,\alpha)|\vargeq N^{% \varepsilon}t_{1}\left(\frac{N^{\omega_{A}}}{N^{\omega_{0}}}+\frac{1}{N^{% \omega_{\ell}}}+\frac{1}{\sqrt{NG}}\right)\right]\varleq N^{-D}, (3.48)

for large enough N.

Proof.  Define w_{i}(t,\alpha):=\hat{z}_{i}(t,\alpha)-\tilde{z}_{i}(t,\alpha). The w_{i} satisfy the equations

 \partial_{t}w_{i}=\sum^{\mathcal{A}_{q_{*}},(i)}_{j}\hat{B}_{ij}(w_{j}-w_{i})+% A_{i} (3.49)

where

 \hat{B}_{ij}=\frac{1}{N}\frac{1}{(\hat{z}_{i}(t,\alpha)-\hat{z}_{j}(t,\alpha))% (\tilde{z}_{i}(t,\alpha)-\tilde{z}_{j}(t,\alpha))}. (3.50)

The error term A_{i} satisfies A_{i}=0 for |i|>N^{\omega_{A}}, and for |i|\varleq N^{\omega_{A}} it is given by

 \displaystyle A_{i} \displaystyle=\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}^{c},(i)}_{j}\frac{1}{\tilde% {z}_{i}(t,\alpha)-\tilde{z}_{j}(t,\alpha)}+\mathrm{Re}[m(\gamma_{0}(t,\alpha),% t,\alpha)] \displaystyle=\left(\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}^{c},(i)}_{j}\frac{1}{% z_{i}(t,\alpha)-z_{j}(t,\alpha)}-\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{% \rho(x,t,\alpha)\mathrm{d}x}{z_{i}(t,\alpha)-x}\right) \displaystyle+\left(\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{\rho(x,t,\alpha)% \mathrm{d}x}{z_{i}(t,\alpha)-x}-\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{\rho% (x,t,\alpha)\mathrm{d}x}{\gamma_{i}(t,\alpha)-x}\right) \displaystyle+\left(\mathrm{Re}[m(\gamma_{0}(t,\alpha),t,\alpha)]-\mathrm{Re}[% m(\gamma_{i}(t,\alpha),t,\alpha)]\right)+\left(\int_{\mathcal{I}_{i}}\frac{% \rho(x,t,\alpha)\mathrm{d}x}{\gamma_{i}(t,\alpha)-x}\right) \displaystyle=:E_{1}+E_{2}+E_{3}+E_{4} (3.51)

The proof of the lemma is as follows. Since both \tilde{z}_{i} and \hat{z}_{i} are ordered the kernel \hat{B}_{ij} are the coefficients of a jump process on [[-(N-1)/2,(N-1)/2]]. Hence the semigroup \mathcal{U}^{\hat{B}} is a contraction on every \ell^{p} space. Since at time t=0 we have \hat{z}_{i}(0,\alpha)=\tilde{z}_{i}(0,\alpha), we have w(t)=\int_{0}^{t}\mathcal{U}^{\hat{B}}(s,t)A(s)\mathrm{d}s by the Duhamel formula. Therefore,

 ||w(t)||_{\infty}\varleq t\sup_{0\varleq s\varleq t}||A(s)||_{\infty}. (3.52)

The remainder of the proof consists of estimating A_{i} using the rigidity estimates.

Let \varepsilon>0. For the remainder of the proof we work on the event that the estimates (3.32) and (3.33) of Section 3.2.1 3 hold with this \varepsilon and a q satisfying q_{*}<q<1, and a small \delta>0 to be determined and large D>0. We take \delta_{1} to satisfy \omega_{0}-\delta_{1}>\omega_{1}.

We fix \eta>N^{2\delta}/N satisfying \eta\ll G. We write the term E_{1} as

 \displaystyle E_{1} \displaystyle=\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}^{c},(i)}_{j}\frac{1}{z_{i}(% t,\alpha)-z_{j}(t,\alpha)}-\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{\rho(x,t,% \alpha)\mathrm{d}x}{z_{i}(t,\alpha)-x} (3.53) \displaystyle=\Bigg{(}\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}^{c},(i)}_{j\in\hat{% \mathcal{C}}_{q}}\frac{1}{z_{i}-z_{j}}-\int_{\hat{\mathcal{I}}_{1}}\frac{\rho(% x,t,\alpha)\mathrm{d}x}{z_{i}-x}\Bigg{)}+\Bigg{(}\frac{1}{N}\sum^{\mathcal{A}_% {q_{*}}^{c},(i)}_{j\notin\hat{\mathcal{C}}_{q}}\left(\frac{1}{z_{i}-z_{j}}-% \frac{1}{z_{i}-z_{j}-\mathrm{i}\eta}\right)\Bigg{)} \displaystyle+\left(\int_{\hat{\mathcal{I}}_{2}}\left(\frac{1}{z_{i}-x-\mathrm% {i}\eta}-\frac{1}{z_{i}-x}\right)\rho(x,t,\alpha)\mathrm{d}x\right)+\left(m_{N% }(z_{i}+\mathrm{i}\eta)-m(z_{i}+\mathrm{i}\eta,t,\alpha)\right) \displaystyle+\Bigg{(}\int_{\hat{\mathcal{I}}_{3}}\frac{\rho(x,t,\alpha)}{z_{i% }-x-\mathrm{i}\eta}-\frac{1}{N}\sum_{j\in\hat{\mathcal{C}}_{q}}\frac{1}{z_{i}-% z_{j}+\mathrm{i}\eta}\Bigg{)} \displaystyle=:F_{1}+F_{2}+F_{3}+F_{4}+F_{5}. (3.54)

Above, the intervals \hat{\mathcal{I}}_{1},\hat{\mathcal{I}}_{2} and \hat{\mathcal{I}}_{3} are defined as follows. They depend on i,\alpha and t but we suppress this for notational simplicity. Let a>0 be the first index not in \hat{\mathcal{C}}_{q}. Define \hat{\mathcal{I}}_{1}:=[\gamma_{-a}(t,\alpha),\gamma_{a}(t,\alpha)]\backslash% \mathcal{I}_{i}(t,\alpha). We define \hat{\mathcal{I}}_{3}:=[\gamma_{-a}(t,\alpha),\gamma_{a}(t,\alpha)]. Lastly, \hat{\mathcal{I}}_{2}:=\hat{\mathcal{I}}_{3}^{c}. Before estimating each of the F_{k} let us explain the motivation for the above decomposition. First we remark that since |i|\varleq N^{\omega_{A}} the interval \mathcal{I}_{i}(t,\alpha) has length |\mathcal{I}_{i}(t,\alpha)|\asymp\ell/N and is contained in [\gamma_{-a}(t,\alpha),\gamma_{a}(t,\alpha)]. Moreover, |\gamma_{\pm a}(t,\alpha)-\gamma_{i}(t,\alpha)|\asymp G. We want to use rigidity to estimate (3.53). However, we only know that rigidity holds for the particles in \hat{\mathcal{C}}_{q}. Hence we break up the terms in (3.53) into two parts. The first is F_{1} which is estimated using rigidity. The remaining particles are distance at least G from \hat{z}_{i}(t,\alpha) and we can use the local law (3.33) on a scale \eta\ll G to estimate this contribution. The terms F_{2}-F_{5} just correspond to some gymnastics to rewrite these particles in a form that can be estimated by the local law (3.33).

The term F_{1} is estimated using rigidity; using (3.32) we easily see that |F_{1}|\varleq CN^{\varepsilon}/N^{\omega_{\ell}}. For F_{2} we use the fact that the restriction |i|\varleq N^{\omega_{A}} and j\notin\hat{\mathcal{C}}_{q} enforces that |z_{i}-z_{j}|\vargeq cG. Since \eta\ll G we may bound F_{2} by

 |F_{2}|\varleq\eta C\sum_{j}\frac{1}{(z_{i}-z_{j})^{2}+cG^{2}}\varleq C\frac{% \eta}{G}\mathrm{Im}\mbox{ }[m_{N}(z_{i}+\mathrm{i}cG)]\varleq C\frac{\eta}{G}. (3.55)

By similar reasoning we get |F_{3}|\varleq C\eta/G. For F_{4} we use the local law estimate (3.33) and get |F_{4}|\varleq N^{\varepsilon}/(N\eta). Lastly for F_{5} we use the optimal rigidity estimate (3.32) and get

 |F_{5}|\varleq\frac{N^{\varepsilon}}{N\eta}C\left(\mathrm{Im}\mbox{ }[m(z_{i}+% \mathrm{i}\eta,t,\alpha)]+\mathrm{Im}\mbox{ }[m_{N}(z_{i}+\mathrm{i}\eta)]% \right)\varleq C\frac{N^{\varepsilon}}{N\eta}. (3.56)

Hence,

 |E_{1}|\varleq CN^{\varepsilon}\left(\frac{1}{N^{\omega_{\ell}}}+\frac{\eta}{G% }+\frac{1}{N\eta}\right)\varleq CN^{\varepsilon}\left(\frac{1}{N^{\omega_{\ell% }}}+\frac{1}{\sqrt{NG}}\right) (3.57)

where we optimized and chose \eta=(G/N)^{1/2} (and chose \delta>0 small enough to allow this choice).

We can bound

 |E_{2}|=\left|\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{\rho(x,t,\alpha)% \mathrm{d}x}{z_{i}-x}-\int_{\mathcal{I}^{c}_{i}(t,\alpha)}\frac{\rho(x,t,% \alpha)\mathrm{d}x}{\gamma_{i}-x}\right|\varleq C\frac{N^{\varepsilon}}{N^{% \omega_{\ell}}}\mathrm{Im}\mbox{ }[m(\gamma_{i}+\mathrm{i}cN^{\omega_{\ell}}/N% ,t,\alpha)]\varleq C\frac{N^{\varepsilon}}{N^{\omega_{\ell}}}. (3.58)

For E_{3} we use (3.17) and get |E_{3}|\varleq CN^{\omega_{A}}/N^{\omega_{0}}. We now have to bound E_{4}. Note that \mathcal{I}_{i} is almost symmetric about \gamma_{i}(t,\alpha). We have

 |\gamma_{i+k}-\gamma_{i}|=|\gamma_{i-k}-\gamma_{i}|\left(1+\mathcal{O}\left(% \frac{k}{Nt_{0}}\right)\right), (3.59)

and so

 |E_{4}|\varleq\left|\int_{\mathcal{I}_{i}}\frac{\rho(x,t,\alpha)-\rho(\gamma_{% i},t,\alpha)}{\gamma_{i}-x}\mathrm{d}x\right|+C\frac{N^{\omega_{\ell}}}{Nt_{0}% }\varleq C\frac{N^{\omega_{\ell}}}{Nt_{0}}. (3.60)

This proves that with overwhelming probability we have

 ||A||_{\infty}\varleq CN^{\varepsilon}\left(\frac{N^{\omega_{A}}}{N^{\omega_{0% }}}+\frac{1}{N^{\omega_{\ell}}}+\frac{1}{\sqrt{NG}}\right). (3.61)

This yields the claim via Duhamel’s formula (3.52). ∎

### 3.4 Derivation of parabolic equation

Define now

 u_{i}:=\partial_{\alpha}\hat{z}_{i}(t,\alpha). (3.62)

The u_{i} satisfy the equation

 \partial_{t}u_{i}=\sum^{\mathcal{A}_{q_{*}},(i)}_{j}B_{ij}(u_{j}-u_{i})+\xi_{i% }=:-(\mathcal{B}u)_{i}+\xi_{i} (3.63)

where

 B_{ij}=\frac{1}{N}\frac{1}{(\hat{z}_{i}-\hat{z}_{j})^{2}} (3.64)

and \xi_{i}=0 for |i|\varleq N^{\omega_{A}} and for |i|>N^{\omega_{A}},

 \xi_{i}=\frac{1}{N}\sum^{\mathcal{A}^{c},(i)}_{j}\frac{\partial_{\alpha}\tilde% {z}_{i}-\partial_{\alpha}\tilde{z}_{j}}{(\tilde{z}_{i}-\tilde{z}_{j})^{2}}+% \partial_{\alpha}\left(\mathrm{Re}[m(t,\gamma_{0}(\alpha,t),\alpha)]\right). (3.65)

Moreover, the initial data is

 u_{i}(0,\alpha)=\alpha z_{i}(0,1)+(1-\alpha)z_{i}(0,0)=\alpha\hat{z}_{i}(0,1)+% (1-\alpha)\hat{z}_{i}(0,0). (3.66)

Note that for \alpha_{1} and \alpha_{2}, the differences \tilde{u}_{i}:=\tilde{z}_{i}(\alpha_{1})-\tilde{z}_{i}(\alpha_{2}) satisfy

 \partial_{t}\tilde{u}_{i}=\frac{1}{N}\sum_{j}\frac{\tilde{u}_{j}-\tilde{u}_{i}% }{(\tilde{z}_{i}(\alpha_{1})-\tilde{z}_{j}(\alpha_{1}))(\tilde{z}_{i}(\alpha_{% 2})-\tilde{z}_{j}(\alpha_{2}))} (3.67)

with \tilde{u}_{i}(0)=(z_{i}(0,1)-z_{i}(0,0))(\alpha_{1}-\alpha_{2}). Since |z_{i}(0,0)|+|z_{i}(0,1)|\varleq N^{C} with overwhelming probability for some C>0 by (3.34) we see that

 ||\partial_{\alpha}\tilde{z}(\alpha,t)||_{\infty}\varleq N^{C} (3.68)

for 0\varleq t\varleq 1 with overwhelming probability. It is not hard to see that

 |\xi_{i}|\varleq\boldsymbol{1}_{\{|i|>N^{\omega_{A}}\}}N^{C} (3.69)

for some C>0 with overwhelming probability.

The parabolic equation (3.63) is the key starting point. We will treat \xi_{i} as an error term. Since it vanishes for indices |i|\varleq N^{\omega_{A}} and the operator \mathcal{B} involves jumps only for particles distance N^{\omega_{\ell}}\ll N^{\omega_{A}} apart we expect that \xi_{i} will have a negligible contribution near 0. This is in fact true as we will see below.

### 3.5 The kernel \mathcal{U}^{B}

At this point essentially the entire remainder of Section 3 and all of Section 4 are concerned only with properties of the semigroup \mathcal{U}^{B} of the kernel \mathcal{B}. The semigroup \mathcal{U}^{B} depends on the \hat{z}_{i}(t,\alpha). The method that we are going to present for analyzing the semigroup \mathcal{U}^{B} is more general and works for any semigroup whose kernel consists of random coefficients satisfying a system of SDEs with certain properties and certain a-priori bounds.

In this section we will pass to a more general set-up involving the hypotheses 1-3 below. The set-up consists of a semigroup \mathcal{U}^{B} and kernel \mathcal{B} with coefficients B_{ij}=N^{-1}(\hat{z}_{i}(t,\alpha)-\hat{z}_{i}(t,\alpha))^{-2}. Here we are abusing notation slightly and re-using \hat{z}_{i}, \mathcal{B}, \mathcal{U}^{B}, etc. In the next few subsections and in Section 4 we will then use these hypotheses to derive various facts about \mathcal{U}^{B}. The main result about \mathcal{U}^{B} is Theorem 3.11 which will be proven at the end of Section 3.6. After proving Theorem 3.11, we will return to the previous set-up and use Theorem 3.11 to prove Theorem 3.7 in Section 3.7.

We let \hat{z}_{i}(t,\alpha) (we will leave in the \alpha notation even though it is unnecessary - for the next few sections \alpha should be regarded as fixed) be the solution to

 \mathrm{d}\hat{z}_{i}(t,\alpha)=\sqrt{\frac{2}{N}}\mathrm{d}B_{i}+\frac{1}{N}% \sum^{\mathcal{A}_{q_{*}},(i)}_{j}\frac{1}{\hat{z}_{i}-\hat{z}_{j}}\mathrm{d}t% +\boldsymbol{1}_{\{|i|\varleq N^{\omega_{A}}\}}F_{i}\mathrm{d}t+\boldsymbol{1}% _{\{|i|>N^{\omega_{A}}\}}J_{i}\mathrm{d}t (3.70)

where F_{i} and J_{i} are adapted bounded processes. The parameters \omega_{A}, \omega_{\ell} and q_{*} are the same as before, and \mathcal{A}_{q_{*}} is defined as above. Previously we also introduced the index k_{0} in the definition of \hat{\mathcal{C}}_{q}:=\{i:|i|\varleq qk_{0}\}. Here, we take k_{0} as a given parameter in the set-up and assume that k_{0}\asymp NG. Let \rho(E,t,\alpha)\mathrm{d}E be measures with densities on |E|\varleq qG for any 0<q<1 (here, we just sent G\to cG in order to simplify notation). Suppose that the following hold.

1. [label=()]

2. We have \rho(0,0,\alpha)=\rho_{\mathrm{sc}}(0) and \gamma_{0}(0,\alpha)=0 and

Moreover, G^{2}\vargeq N^{\sigma}t_{0}=N^{\omega_{0}}/N for some \sigma>0 and \omega_{0}>0. We assume that the classical particle locations \gamma_{i}(\alpha,t) satisfy

 i\in\hat{\mathcal{C}}_{q}\implies\gamma_{i}(\alpha,0)\in[-Gq^{\prime},Gq^{% \prime}] (3.72)

for some 0<q^{\prime}<1 depending on q. We also assume

 |\partial_{t}\gamma_{i}(t,\alpha)|\varleq C\log(N) (3.73)

for i\in\hat{\mathcal{C}}_{q}.

3. We have the rigidity estimate

 \mathbb{P}\left[\sup_{i\in\hat{\mathcal{C}}_{q}}\sup_{0\varleq t\varleq 10t_{1% }}|\hat{z}_{i}(t,\alpha)-\gamma_{i}(t,\alpha)|\vargeq\frac{N^{\varepsilon}}{N}% \right]\varleq N^{-D} (3.74)

for any \varepsilon,D>0 and 0<q<1.

4. For the terms F_{i} and J_{i} we have for some fixed C_{J}>0 and \omega_{F}>0 and every 0<q<1,

 \mathbb{P}\left[\sup_{i\in\hat{\mathcal{C}}_{q}}\sup_{0\varleq t\varleq 10t_{1% }}|J_{i}|\vargeq C_{J}\log(N)\right]\varleq N^{-D} (3.75)

and

 \mathbb{P}\left[\sup_{i}\sup_{0\varleq t\varleq 10t_{1}}|J_{i}|\vargeq N^{C_{J% }}\right]\varleq N^{-D} (3.76)

and

 \mathbb{P}\left[\sup_{i}\sup_{0\varleq t\varleq 10t_{1}}|F_{i}|\vargeq\frac{N^% {\varepsilon}}{N^{\omega_{F}}}\right]\varleq N^{-D} (3.77)

for any \varepsilon,D>0.

Remark. In our case F_{i}=0 but we have added it for the following reason. In our set-up we have F_{i}=0 because we differentiated the short-range approximation to arrive at the parabolic equation (3.63). In other applications it is conceivable that one would like a homogenization result for the full process \mathrm{d}\tilde{z}_{i}. This is covered by the above set-up by rigidity - in this case \omega_{F}=\omega_{\ell} (i.e., the long-range z_{i}-z_{j} terms cancel with \partial_{t}\gamma=-\mathrm{Re}[m(\gamma)]). Finally, while the \omega_{\ell} appearing in the definition of the \hat{z}_{i} and \mathcal{B} are the same, this is not crucial as extra terms can just be absorbed into the F_{i} term using rigidity and the smoothness of the density \rho(E,t,\alpha).

With \hat{z}_{i} satisfying 1-3 we will consider the operator

 (\mathcal{B}u)_{i}:=\sum^{\mathcal{A}_{q_{*}},(i)}_{j}B_{ij}(u_{i}-u_{j}) (3.78)

with semigroup \mathcal{U}^{B}. Before we write down the main result about the semigroup \mathcal{U}^{B} we record some estimates on it. First, we have the following finite speed of propogation estimate.

###### Lemma 3.9.

Let 0\varleq s\varleq t\varleq t_{1}. Let q_{*} and \omega_{\ell} be in the definition of the short-range set \mathcal{A}_{q_{*}} for \mathcal{B}. Suppose that 1-3 hold. Let 0<q_{1}<q_{2}<q_{*} be given. Let D>0 and \varepsilon>0. For each \alpha there is an event \mathcal{F}_{\alpha} with probability \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimates hold. If i\in\hat{\mathcal{C}}_{q_{2}} and 0\varleq s\varleq t\varleq 10t_{1}, then

If i\notin\hat{\mathcal{C}}_{q_{2}} and j\in\hat{\mathcal{C}}_{q_{1}} and 0\varleq s\varleq t\varleq 10t_{1} then

 |\mathcal{U}^{B}_{ji}(s,t,\alpha)|\varleq\frac{1}{N^{D}}. (3.80)

Lemma 3.9 is an immediate consequence of Theorem 4.1. Similar estimates appeared earlier in [18] and our proof follows closely the one appearing there.

Lemma 3.9 contains two estimates. The first (3.79) is almost-optimal in that the kernel decays quickly when |i-j|\gtrsim\ell, where \ell is the range of the jump kernel. Its proof requires the optimal rigidity estimate. We also need the second estimate (3.80) which is weaker but holds for particles i for which rigidity does not hold.

We also have the following estimate for the kernel \mathcal{U}^{B} which says that for the purposes of upper bounds we can think of \mathcal{U}^{B}\sim t/(x^{2}+t^{2}).

###### Lemma 3.10.

Let q_{*} be from the definition of \mathcal{A}_{q_{*}} in the definition of \mathcal{B}. Suppose that 1-3 hold. Let 0<q_{1}<q_{*}, D>0 and \varepsilon>0. For each \alpha there is an event with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimates hold. For i,j\in\hat{\mathcal{C}}_{q_{1}} and 0\varleq s\varleq t\varleq 10t_{1} we have

 \left|\mathcal{U}^{B}_{ij}(s,t)\right|\varleq\frac{N^{\varepsilon}}{N}\frac{|t% -s|\vee N^{-1}}{((i-j)/N)^{2}+(|t-s|\vee N^{-1})^{2}} (3.81)

Remark. Note that the above estimate will not hold for \mathcal{U}^{B} if (s-t)\gg N^{\omega_{\ell}}/N.

The proof of the above estimate is deferred to the next section and is stated there as Theorem 4.7. Roughly, the proof consists of the following steps. First we derive the general estimate

 |\mathcal{U}^{B}_{ij}(0,t)|\varleq\frac{C}{Nt} (3.82)

using the Nash method. This argument is similar to that in [40] - it is slightly different as we only have a short range operator \mathcal{B} living on the scale N^{\omega_{\ell}}, but this does not affect things as long as t\ll N^{\omega_{\ell}}/N. Then we decompose

 \mathcal{B}=\mathcal{S}+\mathcal{R} (3.83)

where \mathcal{S} is a short-range operator on the scale \ell_{2}\sim Nt. (Although \mathcal{B} is already a short-range operator, we are interested in time scales Nt\ll\ell, where \ell is the scale that \mathcal{B} lives on - hence we must make a further long range/short range decomposition of \mathcal{B}). We then prove finite speed estimates for \mathcal{S} and use this together with a Duhamel expansion to derive the estimate.

### 3.6 Homogenization of \mathcal{U}^{B}

In this section we will prove that \mathcal{U}^{B} is given by a deterministic quantity, plus random corrections of lower order. This is the main calculation of Section 3. Fix \varepsilon_{B}>0 s.t.

 \omega_{A}-\varepsilon_{B}>\omega_{\ell}, (3.84)

and let

 |a|\varleq N^{\omega_{A}-\varepsilon_{B}}. (3.85)

We consider a solution w of the equation

Let \mu be the counting measure on [[-(N-1)/2,(N-1)/2]] normalized to have mass 1. We introduce the \ell^{p}-norms

The particle density is smooth on the scale t_{0}. Our operator \mathcal{B} instead lives on the scale N^{\omega_{\ell}}/N\ll t_{0}, and we are working with times t\ll\ell/N\ll t_{0}, and so our solutions w_{i} will never see the density fluctuations. Hence it makes sense to compare w with the solution (on \mathbb{R}) of

 \partial_{t}f(x)=\int_{|x-y|\varleq\eta_{\ell}}\frac{f(y)-f(x)}{(x-y)^{2}}\rho% _{\mathrm{sc}}(0)\mathrm{d}y (3.88)

where

 \eta_{\ell}:=\frac{N^{\omega_{\ell}}}{N\rho_{\mathrm{sc}}(0)}. (3.89)

Let p_{t}(x,y) be the kernel of the above equation. We define the “flat” classical eigenvalue/particle locations by

 \gamma^{(\mathfrak{f})}_{j}:=\frac{j}{N\rho_{\mathrm{sc}}(0)}. (3.90)

Note that for |j|\varleq N^{\omega_{0}/2} we have

 |\gamma^{(\mathfrak{f})}_{j}-\tilde{\gamma}_{j}(t,\alpha)|\varleq\frac{C}{N}. (3.91)

The main result of this section is the following.

###### Theorem 3.11.

Suppose that 1-3 of Section 3.5 hold. Fix an index |a|\varleq N^{\omega_{A}-\varepsilon_{B}}. Let i satisfy |i-a|\varleq\ell/10. Let t_{1} be as above and let

 t_{2}:=N^{-\varepsilon_{2}}t_{1} (3.92)

for \omega_{1}-\varepsilon_{2}>0. Let \varepsilon>0 and D>0. There is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimate holds. For every u with |u|\varleq t_{2} we have

 \left|\mathcal{U}^{B}_{ia}(0,t_{1}+u)-\frac{1}{N}p_{t_{1}}(\gamma^{(\mathfrak{% f})}_{i},\gamma^{(\mathfrak{f})}_{a})\right|\varleq N^{\varepsilon}\frac{N^{% \varepsilon_{2}}}{Nt_{1}}\left\{\frac{(Nt_{1})^{2}}{\ell^{2}}+\frac{1}{(Nt_{1}% )^{1/10}}+\frac{1}{N^{\omega_{F}/3}}\right\}+N^{\varepsilon}\frac{N^{-% \varepsilon_{2}/2}}{Nt_{1}}. (3.93)

In the remainder of Section 3.6 we will work under the assumption that 1-3 hold. The proof of the following lemma is deferred to Section 5.

###### Lemma 3.12.

Let \varepsilon_{1}>0 and D_{1}>0. We have for N^{-D_{1}}\varleq t\varleq N^{-\varepsilon_{1}}\eta_{\ell},

 p_{t}(x,y)\varleq C\frac{t}{(x-y)^{2}+t^{2}}. (3.94)

For any \varepsilon_{2}>0 if |x-y|>N^{\varepsilon_{2}}\eta_{\ell} and N^{-D_{1}}\varleq t\varleq N^{-\varepsilon_{1}}\eta_{\ell},

 p_{t}(x,y)\varleq\frac{1}{N^{D_{2}}} (3.95)

for any D_{2}>0.

For spatial derivatives we have, for N^{-D_{1}}\varleq t\varleq N^{-\varepsilon_{1}}\eta_{\ell},

 p^{(k)}_{t}(x,y)\varleq\frac{C}{t^{k}}p_{t}(x,y)+\frac{1}{N^{D_{2}}}, (3.96)

and

 p^{(k)}_{t}(x,y)\varleq\frac{1}{N^{D}} (3.97)

for any D_{2} if |x-y|>N^{\varepsilon_{2}}\eta_{\ell}.

For the time derivative we have for N^{-D_{1}}\varleq t\varleq N^{-\varepsilon_{1}}\eta_{\ell},

 |\partial_{t}p_{t}(x,y)|\varleq\frac{C}{x^{2}+y^{2}}+N^{-D_{2}}. (3.98)

Remark. The short time cut-off t\vargeq N^{-D_{1}} is technical. In our application we will only take p_{t} with t\vargeq N^{-1}.

We need to introduce two auxilliary scales s_{0} and s_{1}. They will satisfy

 N^{-1}\ll s_{0}\ll s_{1}\ll t_{1}\ll t_{0}. (3.99)

Define now

 f(x,t)=\sum_{j}\frac{1}{N}p_{s_{0}+t-s_{1}}(x,\gamma^{(\mathfrak{f})}_{j})w_{j% }(s_{1}), (3.100)

and

 f_{i}(t):=f(\hat{z}_{i}(t,\alpha),t). (3.101)

We are going to compare w_{i}(t) to f_{i}(t). A more natural choice would perhaps be f_{i}(t)=p_{t}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{a}). We explain here the motivation for the above choice of f_{i} and the introduction of the scales s_{1} and s_{0}. Our method relies on differentiating the \ell^{2} norm of the difference w_{i}-f_{i}, and then integrating it back. We therefore require an estimate on the \ell^{2} norm of the difference at the beginning endpoint of the time interval over which the integration occurs. One choice could be t=0 for the start point of this interval. However, w_{i} is quite singular at this point so we allow it to evolve for a short time s_{1} before comparing f_{i} to w_{i}. At this point one might want to take f(t)=p_{t-s_{1}}\star w(s_{1}). However at t=s_{1}, the kernel p_{t-s_{1}} is a \delta-function and so this convolution operation does not make sense. We therefore introduce the regularizing scale s_{0}. By the standard energy estimate w(s_{1}) has some smoothness on the scale s_{1}. This allows us to control the \ell^{2} distance between w(s_{1}) and its convolution with the approximate \delta-function p_{s_{0}}.

An additional technical complication is that the standard energy estimate involves a time average and so we will have to average the startpoint s_{1} over the interval [s_{1},2s_{1}].

We have the normalization condition

 \sum_{j}\frac{1}{N}p_{t}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})=1+\mathcal{O% }((Nt)^{-1}) (3.102)

and also for \ell_{1}\gg Nt,

 \sum_{|i-j|\varleq\ell_{1}}\frac{1}{N}p_{t}(\hat{z}_{i},\gamma^{(\mathfrak{f})% }_{j})=1+\mathcal{O}((Nt)^{-1})+\mathcal{O}((Nt)/\ell_{1}) (3.103)

and

 \sum_{|i-j|>\ell_{1}}\frac{1}{N}p_{t}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})% \varleq C\frac{Nt}{\ell_{1}} (3.104)

The following lemma provides an estimate on the \ell^{2} norm of the difference w-f. The error is in terms of the scales s_{0},\ell and s_{1} as well as a quantity which can be controlled via the standard energy estimate for w.

###### Lemma 3.13.

Let w be as in (3.86) and f as in (3.100). For any \varepsilon_{1}>0 and \varepsilon_{2}>0 and D>0 there is for each \alpha an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following holds,

 \displaystyle||w(s_{1})-f(s_{1})||_{2}^{2} \displaystyle\varleq \displaystyle s_{0}C\sum_{|i|\varleq N^{\omega_{A}-\varepsilon_{B}}+N^{\omega_% {\ell}+\varepsilon_{2}}}\sum_{|i-j|\varleq\ell}\frac{(w_{i}(s_{1})-w_{j}(s_{1}% ))^{2}}{(i-j)^{2}}+N^{\varepsilon_{1}}\left(\frac{1}{(Ns_{0})^{2}}+\frac{(Ns_{% 0})^{2}}{\ell^{2}}\right)\frac{1}{s_{1}}. (3.105)

Proof.  For notational simplicity let

 {\sum_{i}}^{{}^{\prime}}:=\sum_{|i|\varleq N^{\omega_{A}-\varepsilon_{B}}+N^{% \omega_{\ell}+\varepsilon_{1}}}. (3.106)

We also drop the argument s_{1} and write w_{i}=w_{i}(s_{1}), f_{i}=f_{i}(s_{1}). With overwhelming probability we have,

 \displaystyle\frac{1}{N}\sum_{i}(w_{i}-f_{i})^{2} \displaystyle=\frac{1}{N}{\sum_{i}}^{{}^{\prime}}\left(w_{i}-\sum_{j}\frac{p_{% s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})}{N}w_{j}\right)^{2}+\mathcal{O% }(N^{-D}) \displaystyle\varleq\frac{C}{N}{\sum_{i}}^{{}^{\prime}}\left(\sum_{|j-i|% \varleq\ell}\frac{p_{s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})}{N}(w_{i}% -w_{j})\right)^{2}+C\left(\frac{1}{(Ns_{0})^{2}}+\frac{(Ns_{0})^{2}}{\ell^{2}}% \right)||w||_{2}^{2} (3.107)

In the first line we used the decay estimates from Lemmas 3.9 and 3.12 to change the sum from \sum to \sum^{\prime}. In the inequality we used the normalization condition (3.103), as well as (3.104) which together with Young’s inequality shows that

 \frac{1}{N}\sum_{i}\left(\sum_{|j-i|>\ell}\frac{1}{N}p_{t}(\hat{z}_{i},\gamma^% {(\mathfrak{f})}_{j})w_{j}\right)^{2}\varleq C\frac{(Nt)^{2}}{\ell^{2}}||w||_{% 2}^{2}. (3.108)

We then bound the first term in (3.107) by

 \frac{1}{N}{\sum_{i}}^{{}^{\prime}}\left(\sum_{|j-i|\varleq\ell}\frac{p_{s_{0}% }(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})}{N}(w_{i}-w_{j})\right)^{2}\varleq% \frac{1}{N}{\sum_{i}}^{{}^{\prime}}\left(\sum_{|j-i|\varleq\ell}\frac{p^{2}_{s% _{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})|i-j|^{2}}{N^{2}}\right)\left(% \sum_{|j-i|\varleq\ell}\frac{(w_{i}-w_{j})^{2}}{|i-j|^{2}}\right). (3.109)

We have

 \displaystyle\sum_{j:|j-i|\varleq\ell}\frac{p^{2}_{s_{0}}(x_{i},\gamma^{(% \mathfrak{f})}_{j})|i-j|^{2}}{N^{2}}\varleq CN\int\frac{(s_{0})^{2}x^{2}}{(x^{% 2}+(s_{0})^{2})^{2}}\mathrm{d}x\varleq CNs_{0}. (3.110)

Lastly we can estimate the \ell^{2} norm of w using Lemma 3.10 by

 ||w||_{2}^{2}\varleq\frac{N^{\varepsilon_{1}}}{N}\sum_{i}\frac{(s_{1})^{2}}{((% i-a)/N)^{2}+(s_{1})^{2})^{2}}\varleq\frac{N^{\varepsilon_{1}}}{s_{1}}. (3.111)

These inequalities yield the claim. ∎

The main calculation of the homogenization theorem is the following. We use the Ito lemma to differentiate ||w-f||_{2}^{2}. Roughly what we find is that

 \mathrm{d}||w-f||_{2}^{2}=-||w-f||^{2}_{\dot{H}^{1/2}}\mathrm{d}t+\mbox{lower order} (3.112)

where the lower order terms contains a martingale term as well as other errors{}^{1}222{}^{1} A similar idea was independently discovered in a forthcoming work by Jun Yin and Antti Knowles [51].. Integrating this back gives us control over the homogeneous \dot{H}^{1/2} norm of w-f.

###### Lemma 3.14.

Let w be as in (3.86) and f as in (3.100) with parameters s_{1} and s_{0}. For t\vargeq s_{1} we can write the Ito differential of ||w(t)-f(t)||_{2}^{2} in the form

 \mathrm{d}\frac{1}{N}\sum_{i}(w_{i}-f_{i})^{2}=-\langle w(t)-f(t),\mathcal{B}(% w(t)-f(t))\rangle\mathrm{d}t+X_{t}\mathrm{d}t+\mathrm{d}M_{t} (3.113)

where M_{t} is a martingale and X_{t} is a process implicitly defined by the above equality. We have the following estimates for X_{t} and M_{t}. Let \varepsilon>0 and D>0 be given. For each \alpha there is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimates hold. For s_{1}\varleq t\varleq 9t_{1} we have for X_{t}

 \displaystyle|X_{t}| \displaystyle\varleq\frac{1}{5}\langle w-f,\mathcal{B}(w-f)\rangle \displaystyle+\frac{C}{t+s_{1}}\frac{N^{\varepsilon}}{(t-s_{1}+s_{0})}\bigg{\{% }\frac{1}{\sqrt{N(t-s_{1}+s_{0})}}+\frac{1}{N^{\omega_{F}}}\bigg{\}}. (3.114)

For any u_{1} and u_{2} with 9t_{1}>u_{2}>u_{1}\vargeq s_{1} we have

 \left|\int_{u_{1}}^{u_{2}}\mathrm{d}M_{t}\right|\varleq\frac{N^{\varepsilon}}{% N}\frac{1}{(u_{1}+s_{1})^{3/2}}\frac{1}{(u_{1}-s_{1}+s_{0})^{1/2}}. (3.115)

Proof.  The estimates

and

 f(t,\hat{z}_{i})\varleq\frac{C}{(t-s_{1}+s_{0})} (3.117)

are immediate corollaries of Lemma 3.12. Since ||w(s_{1})||_{\infty}\varleq C(s_{1})^{-1} we obtain

 f(t,\hat{z}_{i})\varleq\frac{C}{t+s_{1}} (3.118)

In the calculations below we implicitly use that for any \varepsilon_{1}>0,

with overwhelming probability by the finite speed estimates of Lemma 3.9 above. For example by our assumption on a this holds for

It will be convenient to use the notation f_{i}^{(k)}:=f^{(k)}(\hat{z}_{i}). It is clear that similar estimates to (3.119) hold for the derivatives f^{\prime}_{i} and f^{\prime\prime}_{i} and (\partial_{t}f)_{i}.

We calculate by the Ito formula,

 \displaystyle d\frac{1}{N}\sum_{i}(w_{i}-f_{i})^{2} \displaystyle= \displaystyle\frac{2}{N}\sum_{i}(w_{i}-f_{i})\left[\partial_{t}w_{i}\mathrm{d}% t-(\partial_{t}f)(t,\hat{z}_{i})\mathrm{d}t-f^{\prime}(t,\hat{z}_{i})\mathrm{d% }\hat{z}_{i}-f^{\prime\prime}(t,\hat{z}_{i})\frac{\mathrm{d}t}{2N}\right]+(f^{% \prime}(t,z_{i}))^{2}\frac{\mathrm{d}t}{2N}. (3.121)

Let us start with the Ito terms. Using (3.116) and (3.118) we obtain

 \left|\frac{1}{N^{2}}\sum_{i}(f^{\prime}_{i})^{2}\right|\varleq\frac{C}{N(t+s_% {0}-s_{1})^{2}}\frac{1}{t+s_{1}}\frac{1}{N}\sum_{i}f_{i}\varleq\frac{C}{N(t+s_% {0}-s_{1})^{2}}\frac{1}{t+s_{1}} (3.122)

and similarly,

 \displaystyle\Big{|}\frac{1}{N^{2}}\sum_{i}(w_{i}-f_{i})f^{\prime\prime}_{i}% \Big{|}\varleq\frac{C}{N(t+s_{0}-s_{1})^{2}}\frac{1}{t+s_{1}}\frac{1}{N}\sum_{% i}w_{i}+f_{i}\varleq\frac{C}{N(t+s_{0}-s_{1})^{2}}\frac{1}{t+s_{1}}. (3.123)

We write

 \displaystyle\frac{1}{N}\sum_{i}(w_{i}-f_{i})(\partial_{t}w_{i}-(\partial_{t}f% )_{i})=\frac{1}{N}\sum_{i}(w_{i}-f_{i})\Big{(}\frac{1}{N}\sum^{\mathcal{A}_{q_% {*}},(i)}_{j}\frac{w_{j}-w_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\frac{1}{N}\sum% ^{\mathcal{A}_{q_{*}},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}% }\Big{)} (3.124) \displaystyle+\frac{1}{N}\sum_{i}(w_{i}-f_{i})\Bigg{(}\frac{1}{N}\sum^{% \mathcal{A}_{q_{*}},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-% \int_{|\hat{z}_{i}-y|\varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_{i% }-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y\Bigg{)} (3.125)

The term (3.124) equals

 \frac{1}{N}\sum_{i}(w_{i}-f_{i})\Bigg{(}\frac{1}{N}\sum^{\mathcal{A}_{q_{*}},(% i)}_{j}\frac{w_{j}-w_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\frac{1}{N}\sum^{% \mathcal{A}_{q_{*}},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}% \Bigg{)}=-\frac{1}{2}\langle w-f,\mathcal{B}(w-f)\rangle. (3.126)

Note that this term is negative and is the first term appearing on the RHS of (3.113). It will be used to account for terms on which we cannot use rigidity. For 0<\omega_{\ell,2}<\omega_{\ell} define

 \displaystyle\mathcal{A}_{2}:=\{(i,j):|i-j|\varleq N^{\omega_{\ell,2}}\}\cup\{% (i,j):ij>0,i,j\notin\hat{\mathcal{C}}_{q_{*}}\}. (3.127)

We write the term (3.125) as

 \displaystyle\frac{1}{N} \displaystyle\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{\mathcal{A}_{q_{*}},(% i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\int_{|\hat{z}_{i}-y|% \varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_{i}-y)^{2}}\rho_{% \mathrm{sc}}(0)\mathrm{d}y\right) \displaystyle=\frac{1}{N}\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{\mathcal{% A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}\right) (3.128) \displaystyle+\frac{1}{N}\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{\mathcal{% A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-% \hat{z}_{j})^{2}}-\int_{|\hat{z}_{i}-y|\varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}% _{i})}{(\hat{z}_{i}-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y\right) (3.129)

We first deal with (3.128). We will later use rigidity to deal with (3.129). Write

 v_{i}:=w_{i}-f_{i}. (3.130)

Using a second order Taylor expansion for f_{i} we have for |i|\varleq N^{\omega_{A}},

 \frac{1}{N}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{j}-\hat{% z}_{i})^{2}}=\frac{1}{N}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{f_{i}^{\prime}}{% \hat{z}_{j}-\hat{z}_{i}}+\mathcal{O}\left(\frac{N^{\omega_{\ell,2}}}{N(t-s_{1}% +s_{0})^{2}}\frac{1}{(t+s_{1})}\right). (3.131)

To estimate the remainder term we used the fact that ||f^{\prime\prime}||_{\infty}\varleq C(t-s_{1}+s_{0})^{-2}(t+s_{1})^{-1} as well as the fact that since |i|\varleq N^{\omega_{A}}, the cardinality of \{j:(j,i)\in\mathcal{A}_{2}\} is less than CN^{\omega_{\ell,2}}. For |i|>N^{\omega_{A}} we just use

 \frac{1}{N}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{j}-\hat{% z}_{i})^{2}}=\frac{1}{N}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{f_{i}^{\prime}}{% \hat{z}_{j}-\hat{z}_{i}}+\mathcal{O}(N^{8}). (3.132)

Using (3.131) and (3.132) and the estimate (3.119) we can write the term (3.128) as

 \displaystyle\frac{1}{N}\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{\mathcal{A% }_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}\right) \displaystyle=\frac{1}{N^{2}}\sum_{i}v_{i}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{% f^{\prime}_{i}}{\hat{z}_{i}-\hat{z}_{j}}+\mathcal{O}\left(\frac{N^{\omega_{% \ell,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}{t+s_{1}}\right). (3.133)

We then write the first term on the RHS of (3.133) as

 \frac{1}{N^{2}}\sum_{i}v_{i}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{f^{\prime}_{i}% }{\hat{z}_{i}-\hat{z}_{j}}=\frac{1}{2}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}% _{2}}\frac{(v_{i}-v_{j})f^{\prime}_{i}+v_{j}(f^{\prime}_{i}-f^{\prime}_{j})}{% \hat{z}_{j}-\hat{z}_{i}}. (3.134)

Using again the estimates (3.119) we bound the second term by

 \displaystyle\left|\frac{1}{2}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}_{2}}% \frac{v_{j}(f^{\prime}_{i}-f^{\prime}_{j})}{\hat{z}_{j}-\hat{z}_{i}}\right| \displaystyle\varleq\left|\frac{1}{2}\frac{1}{N^{2}}\sum_{|i-j|\varleq N^{% \omega_{\ell,2}},|j|\varleq 2N^{\omega_{A}}}\frac{v_{j}(f^{\prime}_{i}-f^{% \prime}_{j})}{\hat{z}_{j}-\hat{z}_{i}}\right|+N^{-D} \displaystyle\varleq C\frac{N^{\omega_{\ell,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}% {t+s_{1}}. (3.135)

The second inequality used (3.116). We bound the first term on the RHS of (3.134) using Schwarz by

 \displaystyle\left|\frac{1}{2}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}_{2}}% \frac{(v_{i}-v_{j})f^{\prime}_{i}}{\hat{z}_{j}-\hat{z}_{i}}\right| \displaystyle\varleq\frac{1}{10}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}_{2}}% \frac{(v_{i}-v_{j})^{2}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}+\frac{C}{N^{2}}\sum_{i% }(f_{i}^{\prime})^{2}\sum^{\mathcal{A}_{2},(i)}_{j}1 \displaystyle\varleq\frac{1}{10}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}_{2}}% \frac{(v_{i}-v_{j})^{2}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}+C\frac{N^{\omega_{\ell% ,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}{t+s_{1}} (3.136)

where we used again the decay estimate (3.119), the estimate for f^{\prime}_{i} and the fact that the cardinality of the set \{j:(i,j)\in\mathcal{A}_{2}\} is bounded by CN^{\omega_{\ell,2}} for |i|\varleq N^{\omega_{A}}. The first term will be absorbed into the \langle(w-f),\mathcal{B}(w-f)\rangle term.

In summary, the estimates (3.133)-(3.136) prove that for the term (3.128) we have

 \displaystyle\left|\frac{1}{N}\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{% \mathcal{A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}% \right)\right|\varleq\frac{1}{10}\frac{1}{N^{2}}\sum_{(i,j)\in\mathcal{A}_{2}}% \frac{(v_{i}-v_{j})^{2}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}+C\frac{N^{\omega_{\ell% ,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}{t+s_{1}}. (3.137)

In order to complete the bound of (3.125) we need to estimate (3.129). This term will be estimated by rigidity. Due to the decay estimates (3.119) we can safely ignore the terms with |i|>N^{\omega_{A}}; i.e., for |i|>N^{\omega_{A}} the term inside the brackets is estimated by

 \left|\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}% \frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\int_{|\hat{z}_{i}-y|\varleq% \eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_{i}-y)^{2}}\rho_{\mathrm{sc}}(% 0)\mathrm{d}y\right|\varleq N^{10}. (3.138)

For the terms with |i|\varleq N^{\omega_{A}} we use the rigidity estimates (3.74) of Section 3.5 2. We write

 \displaystyle\left(\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{% 2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\int_{|y-\hat{z}_% {i}|\varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_{i}-y)^{2}}\rho_{% \mathrm{sc}}(0)\mathrm{d}y\right) \displaystyle= \displaystyle\left(\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{% 2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\int_{\eta_{\ell,% 2}\varleq|y-\hat{z}_{i}|\varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}% _{i}-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y\right) (3.139) \displaystyle- \displaystyle\left(\int_{|y-\hat{z}_{i}|\varleq\eta_{\ell,2}}\frac{f(y)-f(\hat% {z}_{i})}{(\hat{z}_{i}-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y\right), (3.140)

where \eta_{\ell,2}=N^{\omega_{\ell,2}}/(N\rho_{\mathrm{sc}}(0)). The term (3.139) is estimated using rigidity by

 \left|\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}% \frac{f_{j}-f_{i}}{(\hat{z}_{i}-\hat{z}_{j})^{2}}-\int_{\eta_{\ell,2}\varleq|y% -\hat{z}_{i}|\varleq\eta_{\ell}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_{i}-y)^{2}% }\rho_{\mathrm{sc}}(0)\mathrm{d}y\right|\varleq\frac{CN^{\varepsilon}}{N^{% \omega_{\ell,2}}}\frac{1}{t-s_{1}+s_{0}}\frac{1}{t+s_{1}}. (3.141)

The term (3.140) is estimated using a second order Taylor expansion. We write it as

 \int_{|y-\hat{z}_{i}|\varleq\eta_{\ell,2}}\frac{f(y)-f(\hat{z}_{i})}{(\hat{z}_% {i}-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y=\int_{|y-\hat{z}_{i}|\varleq\eta_{% \ell,2}}\frac{f^{\prime}(\hat{z}_{i})}{(\hat{z}_{i}-y)}\rho_{\mathrm{sc}}(0)% \mathrm{d}y+\mathcal{O}\left(\frac{N^{\omega_{\ell,2}}}{N(t-s_{1}+s_{0})^{2}}% \frac{1}{t+s_{1}}\right). (3.142)

We then have that

 \int_{|y-\hat{z}_{i}|\varleq\eta_{\ell,2}}\frac{f^{\prime}(\hat{z}_{i})}{(\hat% {z}_{i}-y)}\rho_{\mathrm{sc}}(0)\mathrm{d}y=0. (3.143)

Combining (3.119) with (3.138) for the terms with |i|>N^{\omega_{A}} and then (3.141) and (3.142) for the remaining terms yields the following estimate for (3.129).

 \displaystyle\left|\frac{1}{N}\sum_{i}(w_{i}-f_{i})\left(\frac{1}{N}\sum^{% \mathcal{A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}\frac{f_{j}-f_{i}}{(\hat{% z}_{i}-\hat{z}_{j})^{2}}-\int_{|y-\hat{z}_{i}|\varleq\eta_{\ell}}\frac{f(y)-f(% \hat{z}_{i})}{(\hat{z}_{i}-y)^{2}}\rho_{\mathrm{sc}}(0)\mathrm{d}y\right)\right| \displaystyle\varleq \displaystyle\frac{CN^{\varepsilon}}{N^{\omega_{\ell,2}}}\frac{1}{t-s_{1}+s_{0% }}\frac{1}{t+s_{1}}+C\frac{N^{\omega_{\ell,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}{% t+s_{1}}. (3.144)

In summary, we see that (3.126) and the estimates (3.137) and (3.6) imply

 \displaystyle\frac{1}{N}\sum_{i}(w_{i}-f_{i})(\partial_{t}w_{i}-(\partial_{t}f% )_{i})=-\frac{1}{2}\langle w-f,\mathcal{B}(w-f)\rangle+Y_{t} (3.145)

where

 |Y_{t}|\varleq\frac{1}{10}\langle w-f,\mathcal{B}(w-f)\rangle+\frac{CN^{% \varepsilon}}{N^{\omega_{\ell,2}}}\frac{1}{t-s_{1}+s_{0}}\frac{1}{t+s_{1}}+C% \frac{N^{\omega_{\ell,2}}}{N(t-s_{1}+s_{0})^{2}}\frac{1}{t+s_{1}}. (3.146)

The remaining term to deal with is

 \displaystyle\frac{1}{N}\sum_{i}(w_{i}-f_{i})f^{\prime}_{i}\mathrm{d}\hat{z}_{i} \displaystyle=\mathrm{d}M_{t}+\frac{1}{N}\sum_{i}(w_{i}-f_{i})f^{\prime}_{i}% \bigg{\{}\frac{1}{N}\sum^{\mathcal{A}_{2},(i)}_{j}\frac{1}{\hat{z}_{i}-\hat{z}% _{j}}+\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}% \frac{1}{\hat{z}_{i}-\hat{z}_{j}} \displaystyle+\boldsymbol{1}_{\{|i|\varleq N^{\omega_{A}}\}}F_{i}+\boldsymbol{% 1}_{\{|i|>N^{\omega_{A}}\}}J_{i}\bigg{\}}\mathrm{d}t. (3.147)

The martingale term is

 dM_{t}=\frac{1}{N}\sum_{i}(w_{i}-f_{i})f^{\prime}_{i}\sqrt{\frac{2}{N}}\mathrm% {d}B_{i} (3.148)

which we estimate later. The first non-martingale term appearing on the RHS of (3.147) is identical to (3.134) (we comment here that they actually appear with the same sign and so do not cancel as one might hope) and so we have, proceeding as above,

 \displaystyle\left|\frac{1}{N}\sum_{i}(w_{i}-f_{i})f^{\prime}_{i}\frac{1}{N}% \sum^{\mathcal{A}_{2},(i)}_{j}\frac{1}{\hat{z}_{i}-\hat{z}_{j}}\right|\varleq% \frac{1}{10}\langle(w-f),\mathcal{B}(w-f)\rangle+C\frac{N^{\omega_{\ell,2}}}{N% (t-s_{1}+s_{0})^{2}}\frac{1}{t+s_{1}}. (3.149)

Using (3.119) and (3.76) we can drop all terms in (3.147) with |i|>N^{\omega_{A}}; i.e., we have with overwhelming probability

 \displaystyle\bigg{|}\sum_{|i|>N^{\omega_{A}}}(w_{i}-f_{i})f_{i}^{\prime}\bigg% {\{}\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{2},(i)}_{j}% \frac{1}{\hat{z}_{i}-\hat{z}_{j}}+J_{i}\bigg{\}}\bigg{|}\varleq\frac{1}{N^{D}}. (3.150)

For the F_{i} terms we have by (3.77) with overwhelming probability,

 \left|\sum_{|i|\varleq N^{\omega_{A}}}(w_{i}-f_{i})f^{\prime}_{i}F_{i}\right|% \varleq C\frac{N^{\varepsilon}}{N^{\omega_{F}}(t-s_{1}+s_{0})(t+s_{1})} (3.151)

For |i|\varleq N^{\omega_{A}}, since

 0=\int_{\eta_{\ell,2}\varleq|y-\hat{z}_{i}|\varleq\eta_{\ell}}\frac{1}{\hat{z}% _{i}-y}\rho_{\mathrm{sc}}(0)\mathrm{d}y, (3.152)

we have by the rigidity estimate (3.74)

 \displaystyle\left|\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_{% 2},(i)}_{j}\frac{1}{\hat{z}_{i}-\hat{z}_{j}}\right| \displaystyle=\left|\frac{1}{N}\sum^{\mathcal{A}_{q_{*}}\backslash\mathcal{A}_% {2},(i)}_{j}\frac{1}{\hat{z}_{i}-\hat{z}_{j}}-\int_{\eta_{\ell,2}\varleq|y-% \hat{z}_{i}|\varleq\eta_{\ell}}\frac{1}{\hat{z}_{i}-y}\rho_{\mathrm{sc}}(0)% \mathrm{d}y\right|\varleq C\frac{N^{\varepsilon}}{N^{\omega_{\ell,2}}}. (3.153)

Hence, ignoring the martingale term (and slightly abusing notation) we have obtained the following bound for (3.147).

 \displaystyle\left|\frac{1}{N}\sum_{i}(w_{i}-f_{i})f_{i}^{\prime}\mathrm{d}% \hat{z}_{i}\right| \displaystyle\varleq\frac{\langle(w-f),\mathcal{B}(w-f)\rangle}{10}+\frac{C}{t% +s_{1}}\frac{N^{\varepsilon}}{(t-s_{1}+s_{0})}\bigg{(}\frac{N^{\omega_{\ell,2}% }}{N(t-s_{1}+s_{0})}+\frac{1}{N^{\omega_{\ell,2}}}+\frac{1}{N^{\omega_{F}}}% \bigg{)} (3.154)

for any \varepsilon>0. The equality (3.113) and estimate (3.14) follow from (3.122), (3.123), (3.145), (3.146) and (3.154), after optimizing and choosing N^{\omega_{\ell,2}}\asymp\sqrt{N(t-s_{1}+s_{0})}.

The quadratic variation of the martingale term satisfies

 \mathrm{d}\langle M\rangle_{t}=\frac{1}{N^{3}}\sum_{i}(w_{i}-f_{i})^{2}(f_{i}^% {\prime})^{2}\mathrm{d}t\varleq\frac{C}{N^{2}}\frac{1}{(t+s_{1})^{3}}\frac{1}{% (t-s_{1}-s_{0})^{2}}\mathrm{d}t (3.155)

with overwhelming probability. Hence by the BDG inequality,

 \displaystyle\mathbb{E}\left[\sup_{u_{2}:9t_{1}\vargeq u_{2}\vargeq u_{1}}% \left|\int_{u_{1}}^{u_{2}}\mathrm{d}M_{t}\right|^{p}\right]\varleq C_{p}\frac{% 1}{N^{p}}\frac{1}{(u_{1}+s_{1})^{3p/2}}\frac{1}{(u_{1}-s_{1}+s_{0})^{p/2}} (3.156)

and so

 \sup_{u_{2}:9t_{1}\vargeq u_{2}\vargeq u_{1}}\left|\int_{u_{1}}^{u_{2}}\mathrm% {d}M_{t}\right|\varleq\frac{N^{\varepsilon}}{N}\frac{1}{(u_{1}+s_{1})^{3/2}}% \frac{1}{(u_{1}-s_{1}+s_{0})^{1/2}} (3.157)

with overwhelming probability. A simple argument using a union bound over u_{1} in a set of cardinality at most N^{2} extends this estimate to all s_{1}<u_{1}<u_{2}<9t_{1}. This yields (3.115). ∎

Lemma 3.14 yields the following corollary, after integration in t. Note that the boundary term at 2t_{1} appears with a negative sign so we can drop it from the RHS of (3.158) below.

###### Corollary 3.15.

Let w be as in (3.86) and f as in (3.100) with parameters s_{1} and s_{0}. Let \varepsilon>0 and D>0. For each \alpha there is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following holds.

 \displaystyle\int_{s_{1}}^{2t_{1}}\langle(w-f),\mathcal{B}(w-f)\mathrm{d}s% \rangle\varleq||(w-f)(s_{1})||_{2}^{2}+\frac{N^{\varepsilon}}{s_{1}}\left\{% \frac{1}{(Ns_{0})^{1/2}}+\frac{1}{N^{\omega_{F}}}\right\} (3.158)

Putting together the last two lemmas yields the following homogenization theorem. It is essentially Theorem 3.11 but with a time average. In the next subsection we will remove the time average.

###### Theorem 3.16.

Let a and i satisfy

For any \varepsilon>0 and D>0 there is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which

 \displaystyle\frac{1}{t_{1}}\int_{0}^{t_{1}}\left(\mathcal{U}^{B}_{ia}(0,t_{1}% +u)-\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}% _{a})\right)^{2}\mathrm{d}u\varleq\frac{N^{\varepsilon}}{(Nt_{1})^{2}}\bigg{\{% }\frac{(Nt_{1})^{4}}{\ell^{4}}+\frac{s_{1}^{2}}{t_{1}^{2}}+\frac{t_{1}}{s_{1}}% \left(\frac{1}{(Ns_{0})^{1/2}}+\frac{1}{N^{\omega_{F}}}+\frac{s_{0}}{s_{1}}% \right)\bigg{\}} (3.160)

Proof.  Define w and f as in (3.86) and (3.100), except replace s_{1} by an auxilliary s_{1}^{\prime}\in[s_{1},2s_{1}]. The reason for doing this is that will eventually have to average s_{1}^{\prime} over [s_{1},2s_{1}]

We estimate for 0\varleq u\varleq t_{1},

 \displaystyle\left(\mathcal{U}^{B}_{ia}(0,t_{1}+u)-\frac{1}{N}p_{t_{1}+u}(% \gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_{a})\right)^{2} \displaystyle\varleq C\left(\frac{1}{N}w_{t_{1}+u}(i)-\frac{1}{N}f_{t_{1}+u}(i% )\right)^{2} (3.161) \displaystyle+C\left(\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma% ^{(\mathfrak{f})}_{a})-\frac{1}{N}p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i}% ,\gamma^{(\mathfrak{f})}_{a})\right)^{2} (3.162) \displaystyle+C\left(\frac{1}{N}p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},% \gamma^{(\mathfrak{f})}_{a})-\frac{1}{N}f_{t_{1}+u}(i)\right)^{2} (3.163)

The terms (3.162) and (3.163) are estimated using essentially the regularity of p_{t}(x,y). The remaining term (3.161) is estimated using the last corollary.

We can estimate the term (3.162) using the results from Lemma 3.12 and the optimal rigidity estimate (3.74) from Section 3.5 2. We obtain,

 \displaystyle\left(\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma^{% (\mathfrak{f})}_{a})-\frac{1}{N}p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},% \gamma^{(\mathfrak{f})}_{a})\right)^{2}\varleq C\left(\frac{1}{N}p_{t_{1}+u}(% \gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_{a})-\frac{1}{N}p_{t_{1}+u% }(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{a})\right)^{2} \displaystyle+ \displaystyle C\left(\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma% ^{(\mathfrak{f})}_{a})-\frac{1}{N}p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\gamma^{(% \mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_{a})\right)^{2}\varleq\frac{N^{% \varepsilon}}{(Nt_{1})^{4}}+\frac{C}{(Nt_{1})^{2}}\frac{s_{1}^{2}}{t_{1}^{2}}. (3.164)

For (3.163) we have, using the normalization N^{-1}\sum w(j)=1, the profile from Lemma 3.10, and the estimates from Lemma 3.12,

 \displaystyle\left(\frac{1}{N}p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},% \gamma^{(\mathfrak{f})}_{a})-\frac{1}{N}f_{t_{1}+u}(i)\right)^{2} \displaystyle= \displaystyle\left(\frac{1}{N^{2}}\sum_{j}w_{s_{1}^{\prime}}(j)\left(p_{t_{1}+% u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{a})-p_{t_{1}+u-s_% {1}^{\prime}+s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})\right)\right)^{2} \displaystyle\varleq \displaystyle N^{\varepsilon}\left(\frac{1}{N^{2}}\sum_{j}\frac{s_{1}}{(|j-a|/% N)^{2}+s_{1}^{2}}\frac{1}{t_{1}}\left[\left(\frac{|j-a|}{Nt_{1}}\right)\wedge 1% \right]\right)^{2} \displaystyle\varleq \displaystyle CN^{\varepsilon}\left(\frac{1}{Nt_{1}}\frac{1}{t_{1}}\int_{|x|% \varleq t_{1}}\frac{s_{1}|x|}{x^{2}+s_{1}^{2}}\mathrm{d}x\right)^{2}+CN^{% \varepsilon}\left(\frac{1}{Nt_{1}}\int_{|x|>t_{1}}\frac{s_{1}}{s_{1}^{2}+x^{2}% }\mathrm{d}x\right)^{2}\varleq\frac{CN^{2\varepsilon}}{(Nt_{1})^{2}}\frac{s_{1% }^{2}}{t_{1}^{2}}. (3.165)

Above, we used the fact that

 \left|p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{a}% )-p_{t_{1}+u-s_{1}^{\prime}+s_{0}}(\hat{z}_{i},\gamma^{(\mathfrak{f})}_{j})% \right|\varleq\frac{C}{t_{1}}\min\left\{\frac{|j-a|}{Nt_{1}},1\right\}. (3.166)

We now deal with (3.161). We estimate

 \displaystyle\left(\frac{1}{N}w_{t_{1}+u}(i)-\frac{1}{N}f_{t_{1}+u}(i)\right)^% {2} \displaystyle\varleq \displaystyle C\left(\frac{1}{N}w_{t_{1}+u}(i)-\frac{1}{N}\frac{1}{\ell}\sum_{% |j-i|\varleq\ell}w_{t_{1}+u}(j)-\frac{1}{N}f_{t_{1}+u}(i)+\frac{1}{N}\frac{1}{% \ell}\sum_{|j-i|\varleq\ell}f_{t_{1}+u}(j)\right)^{2} \displaystyle+ \displaystyle C\left(\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}w_{t_{1}+% u}(j)-\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}f_{t_{1}+u}(j)\right)^{2} (3.167)

We apply the Sobolev inequality of Lemma D.1 to the difference of the sequences on \{k:|i-k|\varleq\ell\} (with the N in Lemma D.1 being \ell)

 \left\{w_{t_{1}+u}(k)-\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}w_{t_{1}% +u}(j)\right\}_{k},\left\{f_{t_{1}+u}(k)-\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|% \varleq\ell}f_{t_{1}+u}(j)\right\}_{k} (3.168)

which now have mean 0. We find,

 \displaystyle\left(\frac{1}{N}w_{t_{1}+u}(i)-\frac{1}{N}f_{t_{1}+u}(i)\right)^% {2}\varleq \displaystyle\frac{N^{\varepsilon}}{N^{2}}\langle(w-f)(t_{1}+u),\mathcal{B}(w-% f)(t_{1}+u)\rangle \displaystyle+ \displaystyle C\left(\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}w_{t_{1}+% u}(j)-\frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}f_{t_{1}+u}(j)\right)^{2}. (3.169)

We used the fact that |z_{i}-z_{j}|\vargeq N^{-\varepsilon/4}|i-j|/N. We have

 \frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}w_{t_{1}+u}(j)=\frac{1}{\ell}+% \mathcal{O}\left(\frac{Nt_{1}}{\ell^{2}}\right). (3.170)

Similarly, (using (3.102)) we have

 \frac{1}{N}\frac{1}{\ell}\sum_{|j-i|\varleq\ell}f_{t_{1}+u}(j)=\frac{1}{\ell}+% \mathcal{O}\left(\frac{1}{\ell Nt_{1}}+\frac{Nt_{1}}{\ell^{2}}\right). (3.171)

Hence,

 \displaystyle\left(\frac{1}{N}w_{t_{1}+u}(i)-\frac{1}{N}f_{t_{1}+u}(i)\right)^% {2}\varleq \displaystyle\frac{N^{\varepsilon}}{N^{2}}\langle(w-f)(t_{1}+u),\mathcal{B}(w-% f)(t_{1}+u)\rangle+C\frac{1}{\ell^{2}(Nt_{1})^{2}}+C\frac{(Nt_{1})^{2}}{\ell^{% 4}}

We now apply Corollary 3.15 to obtain that there is an event with overwhelming probability (which depends on the choice of s_{1}^{\prime}), such that

 \displaystyle\int_{s_{1}^{\prime}}^{2t_{1}}\langle(w-f),\mathcal{B}(w-f)% \rangle\mathrm{d}s\varleq\frac{N^{\varepsilon}}{s_{1}}\left(\frac{1}{(Ns_{0})^% {1/2}}+\frac{1}{N^{\omega_{F}}}\right)+||(w-f)(s_{1}^{\prime})||_{2}^{2}. (3.172)

We can average over s_{1}^{\prime}\in[s_{1},2s_{1}] (even though the event described above is s_{1}^{\prime}-dependent, since each holds with overwhelming probability and \mathcal{U}^{B} and p_{t} are bounded, we can apply Lemma E.1) and obtain that with overwhelming probability,

 \displaystyle\frac{1}{t_{1}}\int_{0}^{t_{1}}\left(\mathcal{U}^{B}_{t_{1}+u}(i,% a)-\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_% {a})\right)^{2}\mathrm{d}u\varleq\frac{1}{N^{2}t_{1}s_{1}}\int_{0}^{s_{1}}||(w% -f)(s_{1}+u)||_{2}^{2}\mathrm{d}u \displaystyle+ \displaystyle\frac{CN^{\varepsilon}}{(Nt_{1})^{2}}\left(\frac{1}{(Nt_{1})^{2}}% +\frac{s_{1}^{2}}{t_{1}^{2}}+\frac{1}{\ell^{2}}+\frac{(Nt_{1})^{4}}{\ell^{4}}+% \frac{t_{1}}{s_{1}(Ns_{0})^{1/2}}+\frac{t_{1}}{s_{1}N^{\omega_{F}}}\right). (3.173)

Note that on the RHS the choice of f itself has an s_{1}+u dependence. By Lemma 3.13 we have

 \displaystyle\int_{0}^{s_{1}}||(w-f)(s_{1}+u)||_{2}^{2}\mathrm{d}u\varleq N^{% \varepsilon}\left(\frac{1}{(Ns_{0})^{2}}+\frac{(Ns_{0})^{2}}{\ell^{2}}\right) \displaystyle+ \displaystyle Cs_{0}\int_{0}^{s_{1}}\sum_{|i|\varleq N^{\omega_{A}}}\sum_{|i-j% |\varleq\ell}\frac{(w_{i}(s_{1}+u)-w_{j}(s_{1}+u))^{2}}{(i-j)^{2}}\mathrm{d}u. (3.174)

With overwhelming probability we have

 \displaystyle\int_{0}^{s_{1}}\sum_{\begin{subarray}{c}|i|\varleq N^{\omega_{A}% }\\ |i-j|\varleq\ell\end{subarray}}\frac{(w_{i}(s_{1}+u)-w_{j}(s_{1}+u))^{2}}{(i-j% )^{2}}\mathrm{d}u \displaystyle\varleq N^{\varepsilon}\int_{0}^{s_{1}}\langle w,\mathcal{B}w(s_{% 1}+u)\rangle\mathrm{d}u\varleq N^{\varepsilon}||w(s_{1})||_{2}^{2}\varleq\frac% {CN^{2\varepsilon}}{s_{1}} (3.175)

where in the second inequality we used the standard energy estimate \partial_{t}||w||_{2}^{2}=-\langle w,\mathcal{B}w\rangle. The claim now follows after simplifying the errors. In particular we use,

#### 3.6.1 Removal of time average

Let \varepsilon_{2}>0 and let

 t_{2}:=t_{1}N^{-\varepsilon_{2}}. (3.177)

In this section we show how to remove the time average in Theorem 3.16. More precisely, we prove the following theorem. It is deduced from Theorem 3.16 using only the fact that \mathcal{U}^{B} is a semigroup, the decay properties of \mathcal{U}^{B} given by Lemma 3.10 and the regularity of p_{t}(x,y).

###### Theorem 3.17.

Let a satisfy

 |a|\varleq N^{\omega_{A}-\varepsilon_{B}}/2 (3.178)

and i satisfy

 |i-a|\varleq\frac{\ell}{20}. (3.179)

For any \varepsilon>0 and D>0 there is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which

 \displaystyle\left|\mathcal{U}^{B}_{ia}(0,t_{1}+2t_{2})-\frac{1}{N}p_{t_{1}}(% \gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_{a})\right| \displaystyle\varleq \displaystyle CN^{\varepsilon}\frac{N^{\varepsilon_{2}}}{Nt_{1}}\bigg{\{}\frac% {s_{1}^{2}}{t_{1}^{2}}+\frac{(Nt_{1})^{4}}{\ell^{4}}+\frac{t_{1}}{s_{1}}\left(% \frac{1}{(Ns_{0})^{1/2}}+\frac{1}{N^{\omega_{F}}}+\frac{s_{0}}{s_{1}}\right)% \bigg{\}}^{1/2} \displaystyle+ \displaystyle\frac{N^{\varepsilon}}{Nt_{1}}N^{-\varepsilon_{2}/2}. (3.180)

Proof.  Theorem 3.16 implies that we have with overwhelming probability

 \displaystyle\frac{1}{t_{2}}\int_{0}^{t_{2}}\left|\mathcal{U}^{B}_{jk}(0,t_{1}% +u)-\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{j},\gamma^{(\mathfrak{f})}% _{k})\right|\mathrm{d}u \displaystyle\varleq \displaystyle N^{\varepsilon}\frac{N^{\varepsilon_{2}}}{Nt_{1}}\bigg{\{}\frac{% s_{1}^{2}}{t_{1}^{2}}+\frac{(Nt_{1})^{4}}{\ell^{4}}+\frac{t_{1}}{s_{1}}\left(% \frac{1}{(Ns_{0})^{1/2}}+\frac{1}{N^{\omega_{F}}}+\frac{s_{0}}{s_{1}}\right)% \bigg{\}}^{1/2}, (3.181)

for k\varleq N^{\omega_{A}-\varepsilon_{B}} and |j-k|\varleq\ell/10. For notational simplicity let us denote

 \Phi:=N^{\varepsilon_{2}}\bigg{\{}\frac{s_{1}^{2}}{t_{1}^{2}}+\frac{(Nt_{1})^{% 4}}{\ell^{4}}+\frac{t_{1}}{s_{1}}\left(\frac{1}{(Ns_{0})^{1/2}}+\frac{1}{N^{% \omega_{F}}}+\frac{s_{0}}{s_{1}}\right)\bigg{\}}^{1/2} (3.182)

By the semigroup property we can write for any 0\varleq u\varleq t_{2},

 \mathcal{U}^{B}_{ai}(0,t_{1}+2t_{2})=\sum_{j}\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1% }+2t_{2})\mathcal{U}^{B}_{ji}(0,t_{1}+u) (3.183)

and so we can take an average over u and obtain

 \mathcal{U}^{B}_{ai}(0,t_{1}+2t_{2})=\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}% \mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})\mathcal{U}^{B}_{ji}(0,t_{1}+u)% \mathrm{d}u. (3.184)

We now rewrite the RHS as

 \displaystyle\mathcal{U}^{B}_{ai}(0,t_{1}+2t_{2}) \displaystyle=\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}(t_{1% }+u,t_{1}+2t_{2})\left(\mathcal{U}^{B}_{ji}(0,t_{1}+u)-\frac{1}{N}p_{t_{1}+u}(% \gamma^{(\mathfrak{f})}_{j},\gamma^{(\mathfrak{f})}_{i})\right)\mathrm{d}u (3.185) \displaystyle+\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}(t_{1% }+u,t_{1}+2t_{2})\frac{1}{N}\left(p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{j},% \gamma^{(\mathfrak{f})}_{i})-p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{a},\gamma^{(% \mathfrak{f})}_{i})\right)\mathrm{d}u (3.186) \displaystyle+\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}(t_{1% }+u,t_{1}+2t_{2})\left(\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{a},% \gamma^{(\mathfrak{f})}_{i})\right)\mathrm{d}u (3.187)

From Lemma 3.10, we have the estimate

 |\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})|\varleq\frac{1}{N}\frac{N^{% \varepsilon}t_{2}}{((a-j)/N)^{2}+t_{2}^{2}} (3.188)

from which we see that for any \delta>0,

 \sum_{j:|j-a|>Nt_{2}N^{\delta}}|\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})|% \varleq\frac{N^{\varepsilon}}{N^{\delta}} (3.189)

and also

 \sum_{j:|j-a|\varleq Nt_{2}N^{\delta}}\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2% })=1+N^{\varepsilon}\mathcal{O}\left(\frac{1}{N^{\delta}}\right). (3.190)

Fix a \delta>0 s.t. \delta<\varepsilon_{2}. We also have the estimate

 \left|\mathcal{U}^{B}_{ji}(0,t_{1}+u)\right|+\left|\frac{1}{N}p_{t_{1}+u}(% \gamma^{(\mathfrak{f})}_{j},\gamma^{(\mathfrak{f})}_{i})\right|\varleq\frac{N^% {\varepsilon}}{Nt_{1}}. (3.191)

We use these to estimate the term (3.185) by

 \displaystyle\left|\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}% (t_{1}+u,t_{1}+2t_{2})\left(\mathcal{U}^{B}_{ji}(0,t_{1}+u)-\frac{1}{N}p_{t_{1% }+u}(\gamma^{(\mathfrak{f})}_{j},\gamma^{(\mathfrak{f})}_{i})\right)\mathrm{d}% u\right| \displaystyle\varleq \displaystyle\sum_{j:|j-a|>Nt_{2}N^{\delta}}\frac{1}{t_{2}}\int_{0}^{t_{2}}% \mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})\frac{N^{\varepsilon}}{Nt_{1}}% \mathrm{d}u \displaystyle+ \displaystyle\sum_{j:|j-a|\varleq Nt_{2}N^{\delta}}\frac{1}{N}\frac{N^{% \varepsilon}t_{2}}{((j-a)/N)^{2}+t_{2}^{2}}\frac{1}{t_{2}}\int_{0}^{t_{2}}% \left|\mathcal{U}^{B}_{ji}(0,t_{1}+u)-\frac{1}{N}p_{t_{1}+u}(\gamma^{(% \mathfrak{f})}_{j},\gamma^{(\mathfrak{f})}_{i})\right|\mathrm{d}u \displaystyle\varleq \displaystyle\frac{N^{2\varepsilon}}{Nt_{1}N^{\delta}}+\frac{N^{2\varepsilon}}% {Nt_{1}}\Phi. (3.192)

Note that we are allowed to apply the estimate (3.6.1) because |j-a|\varleq Nt_{2}N^{\delta}\ll\ell which implies |j-i|<\ell/10. For |j-a|\varleq N^{\delta}(Nt_{2}) we have the estimate

 \frac{1}{N}\left|p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{j},\gamma^{(\mathfrak{f}% )}_{i})-p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{a},\gamma^{(\mathfrak{f})}_{i})% \right|\varleq N^{\varepsilon}\frac{1}{Nt_{1}}\frac{Nt_{2}N^{\delta}}{Nt_{1}}. (3.193)

Therefore we can estimate (3.186) by

 \displaystyle\left|\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}% (t_{1}+u,t_{1}+2t_{2})\frac{1}{N}\left(p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{j}% ,\gamma^{(\mathfrak{f})}_{i})-p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{a},\gamma^{% (\mathfrak{f})}_{i})\right)\mathrm{d}u\right| \displaystyle\varleq \displaystyle\sum_{j:|j-a|>Nt_{2}N^{\delta}}\frac{1}{t_{2}}\int_{0}^{t_{2}}% \mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})\frac{N^{\varepsilon}}{Nt_{1}}% \mathrm{d}u \displaystyle+ \displaystyle\sum_{j:|j-a|\varleq Nt_{2}N^{\delta}}\frac{1}{t_{2}}\int_{0}^{t_% {2}}\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})N^{\varepsilon}\frac{1}{Nt_{1}}% \frac{Nt_{2}N^{\delta}}{Nt_{1}}\mathrm{d}u \displaystyle\varleq N^{2\varepsilon}\frac{1}{Nt_{1}}\left(\frac{1}{N^{\delta}% }+\frac{N^{\delta}}{N^{\varepsilon_{2}}}\right). (3.194)

Lastly, since \sum_{j}\mathcal{U}^{B}_{aj}(t_{1}+u,t_{1}+2t_{2})=1 we get

 \displaystyle\sum_{j}\frac{1}{t_{2}}\int_{0}^{t_{2}}\mathcal{U}^{B}_{aj}(t_{1}% +u,t_{1}+2t_{2})\left(\frac{1}{N}p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{a},% \gamma^{(\mathfrak{f})}_{i})\right)\mathrm{d}u \displaystyle=\frac{1}{t_{2}}\int_{0}^{t_{2}}\left(\frac{1}{N}p_{t_{1}+u}(% \gamma^{(\mathfrak{f})}_{a},\gamma^{(\mathfrak{f})}_{i})\right) \displaystyle=\frac{1}{N}p_{t_{1}+2t_{2}}(\gamma^{(\mathfrak{f})}_{a},\gamma^{% (\mathfrak{f})}_{i})+\frac{1}{Nt_{1}}\mathcal{O}\left(N^{-\varepsilon_{2}}% \right). (3.195)

Here we used

 \frac{1}{N}\left|p_{t_{1}+u}(\gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f}% )}_{a})-p_{t_{1}}(\gamma^{(\mathfrak{f})}_{i},\gamma^{(\mathfrak{f})}_{a})% \right|\varleq C\frac{N^{-\varepsilon_{2}}}{Nt_{1}}. (3.196)

This yields the claim after taking \delta=\varepsilon_{2}/2. ∎

At this point we just have to choose the parameters s_{0} and s_{1} to conclude the homogenization result for \mathcal{U}^{B}.

Proof of Theorem 3.11. We use the result of Theorem 3.17 and just make a choice of s_{0} and s_{1}. First we optimize over s_{0} and take

 (Ns_{0})=(Ns_{1})^{2/3}. (3.197)

The error then simplifies to

 N^{\varepsilon}\frac{N^{\varepsilon_{2}}}{Nt_{1}}\left\{\frac{s_{1}^{2}}{t_{1}% ^{2}}+\frac{(Nt_{1})^{4}}{\ell^{4}}+\frac{t_{1}}{s_{1}}\left(\frac{1}{N^{% \omega_{F}}}+\frac{1}{(Ns_{1})^{1/3}}\right)\right\}^{1/2}+N^{\varepsilon}% \frac{N^{-\varepsilon_{2}/2}}{Nt_{1}}. (3.198)

To optimize over s_{1} we have two cases. If \omega_{F}\vargeq\omega_{1}3/10 then we take Ns_{1}=(Nt_{1})^{9/10}. If \omega_{F}\varleq\omega_{1}3/10 then we take Ns_{1}=Nt_{1}N^{-\omega_{F}/3}. The error simplifies to

 N^{\varepsilon}\frac{N^{\varepsilon_{2}}}{Nt_{1}}\left\{\frac{(Nt_{1})^{4}}{% \ell^{4}}+\frac{1}{(Nt_{1})^{1/5}}+\frac{1}{N^{\omega_{F}2/3}}\right\}^{1/2}+N% ^{\varepsilon}\frac{N^{-\varepsilon_{2}/2}}{Nt_{1}}. (3.199)

This is the claim. ∎

### 3.7 Completion of proof of Theorem 3.7

First of all we see by the definitions of \tilde{z}_{i} and \hat{z}_{i} (see (3.40) for the former and (3.45)-(3.3) for the latter) and Lemma 3.8 that with overwhelming probability,

 \displaystyle z_{i}(t_{1}+u,1)-z_{i}(t_{1}+u,0) \displaystyle=\left(\tilde{z}_{i}(t_{1}+u,1)-\tilde{z}_{i}(t_{1}+u,0)\right)+% \left(\gamma_{0}(t_{1}+u,1)-\gamma_{0}(t_{1}+u,0)\right) \displaystyle=\left(\hat{z}_{i}(t_{1}+u,1)-\hat{z}_{i}(t_{1}+u,0)\right)+\left% (\gamma_{0}(t_{1}+u,1)-\gamma_{0}(t_{1}+u,0)\right) \displaystyle+\mathcal{O}\left(N^{\varepsilon}t_{1}\left(\frac{N^{\omega_{A}}}% {N^{\omega_{0}}}+\frac{1}{N^{\omega_{\ell}}}+\frac{1}{\sqrt{NG}}\right)\right). (3.200)

Note that the first equation is just by definition - the classical particle locations are defined together with \tilde{z}_{i} at the start of Section 3.3.

With u_{i}=\partial_{\alpha}\hat{z}_{i} we have

 \left(\hat{z}_{i}(t_{1}+u,1)-\hat{z}_{i}(t_{1}+u,0)\right)=\int_{0}^{1}u_{i}(t% _{1}+u,\alpha)\mathrm{d}\alpha. (3.201)

Recall u satisfies \partial_{t}u(\alpha)=\mathcal{B}(\alpha)u(\alpha)+\xi(\alpha) with \xi(\alpha) defined as in Section 3.4, and initial data

 u_{i}(0,\alpha)=\alpha\hat{z}_{i}(0,1)+(1-\alpha)\hat{z}_{i}(0,0). (3.202)

By the bound (3.69), the assumption (3.34) and Lemma 3.9 we see that for any small \delta_{B}>0

 \sup_{|i|\varleq N^{\omega_{A}-\delta_{B}},|u|\varleq t_{1}}\left|u_{i}(t_{1}+% u,\alpha)-v_{i}(t_{1}+u,\alpha)\right|\varleq\frac{1}{N^{10}} (3.203)

with overwhelming probability where v_{i}(\alpha) is defined by

Fix an \varepsilon_{a}>0 and consider the solution

Since |u_{i}(0,\alpha)|\varleq N^{\varepsilon}/N for any |i|\varleq N^{\omega_{0}/2} with overwhelming probability, we see by Lemma 3.10 that for |i|\varleq Nt_{1}N^{\varepsilon_{b}} with \varepsilon_{b}<\varepsilon_{a},

 \displaystyle\left|v_{i}(t_{1}+u,\alpha)-w_{i}(t_{1}+u,\alpha)\right| \displaystyle\varleq\left|\sum_{N^{\omega_{1}+\varepsilon_{a}}<|j|\varleq N^{% \omega_{A}}}\mathcal{U}^{B}_{ij}(0,t_{1}+u,\alpha)u_{j}(0,\alpha)\right| \displaystyle\varleq\frac{N^{\varepsilon}}{N}Nt_{1}\sum_{|j|>N^{\omega_{1}+% \varepsilon_{a}}}\frac{1}{(i-j)^{2}}\varleq C\frac{N^{\varepsilon}}{NN^{% \varepsilon_{a}}}. (3.206)

Therefore,

 \displaystyle z_{i}(t_{1}+u,1)-z_{i}(t_{1}+u,0) \displaystyle=\left(\gamma_{0}(t_{1}+u,1)-\gamma_{0}(t_{1}+u,0)\right) \displaystyle+\int_{0}^{1}\sum_{|j|\varleq Nt_{1}N^{\varepsilon_{a}}}\mathcal{% U}^{B}_{ij}(0,t_{1}+u,\alpha)(z_{j}(0,1)-z_{j}(0,0))\mathrm{d}\alpha \displaystyle+\frac{N^{\varepsilon}}{N}\mathcal{O}\left(N^{\omega_{1}}\left(% \frac{N^{\omega_{A}}}{N^{\omega_{0}}}+\frac{1}{N^{\omega_{\ell}}}+\frac{1}{% \sqrt{NG}}\right)+\frac{1}{N^{\varepsilon_{a}}}\right) (3.207)

with overwhelming probability for |i|\varleq Nt_{1}N^{\varepsilon_{b}}. Note that we used Lemma E.1, and that u(\alpha),v(\alpha),w(\alpha) are all bounded by N^{C} on an \alpha-independent event of overwhelming probability. Theorem 3.7 now follows from an application of Theorem 3.11 with \omega_{F}=\infty. ∎

### 3.8 Proof of Theorem 3.1

Theorem 3.7 implies, after re-writing z_{i}(t,1) in terms of x_{i}, that with overwhelming probability we have,

 \displaystyle\left(x_{i_{0}+i}(t_{0}+t_{1}+u)-\gamma_{i_{0}}(t_{0}+t_{1}+u)% \right)-y_{N/2+i}(t_{0}+t_{1}+u) \displaystyle= \displaystyle\sum_{|j|\varleq Nt_{1}N^{\varepsilon_{a}}}\zeta(N^{-1}(i-j),t_{1% })\left[\left(x_{i_{0}+j}(t_{0}+t_{1})-\gamma_{i_{0}}(t_{0}+t_{1})\right)-y_{N% /2+j}(t_{0}+t_{1})\right] \displaystyle+ \displaystyle\frac{N^{\varepsilon}}{N}\mathcal{O}\left(N^{\omega_{1}}\left(% \frac{N^{\omega_{A}}}{N^{\omega_{0}}}+\frac{1}{N^{\omega_{\ell}}}+\frac{1}{% \sqrt{NG}}\right)+\frac{1}{N^{\varepsilon_{a}}}+N^{\varepsilon_{2}+\varepsilon% _{a}}\left(\frac{(Nt_{1})^{2}}{\ell^{2}}+\frac{1}{(Nt_{1})^{1/10}}\right)+N^{% \varepsilon_{a}-\varepsilon_{2}/2}+\frac{N^{\omega_{1}}}{N^{\omega_{0}/2}}% \right), (3.208)

for any |i|\varleq Nt_{1}N^{\varepsilon_{b}} and |u|\varleq t_{2}. Recall that \zeta was defined in (3.7) (and then p_{t} as the fundamental solution to (3.88)). We also used Lemma 3.4 to replace the classical particle locations from the interpolating measures with those coming from the original free convolution \rho_{\mathrm{fc},t} and the semicircle law. We choose \omega_{A}=(\omega_{\ell}+\omega_{0}/2)/2. The error simplifies to

 \frac{N^{2\varepsilon}}{N}\mathcal{O}\left(N^{\omega_{1}}\left(\frac{1}{N^{% \omega_{\ell}}}\right)+\frac{1}{N^{\varepsilon_{a}}}+N^{\varepsilon_{2}+% \varepsilon_{a}}\left(\frac{(Nt_{1})^{2}}{\ell^{2}}+\frac{1}{(Nt_{1})^{1/10}}% \right)+N^{\varepsilon_{a}-\varepsilon_{2}/2}\right). (3.209)

There are two cases. First if \omega_{1}\vargeq 10\omega_{0}/21 then the error simplifies to

 \frac{N^{2\varepsilon}}{N}\mathcal{O}\left(N^{\omega_{1}}\left(\frac{1}{N^{% \omega_{\ell}}}\right)+\frac{1}{N^{\varepsilon_{a}}}+N^{\varepsilon_{2}+% \varepsilon_{a}}\left(\frac{(Nt_{1})^{2}}{\ell^{2}}\right)+N^{\varepsilon_{a}-% \varepsilon_{2}/2}\right). (3.210)

In this case we then take \varepsilon_{2}=4(\omega_{\ell}-\omega_{1})/3 and then \varepsilon_{a}=(\omega_{\ell}-\omega_{1})/3. The error simplifies to

 \frac{N^{3\varepsilon}}{N}\mathcal{O}\left(\frac{N^{\omega_{1}/3}}{N^{\omega_{% \ell}/3}}\right). (3.211)

We then take \omega_{\ell}=\omega_{0}/2-\varepsilon and so the error is

 \frac{N^{4\varepsilon}}{N}\mathcal{O}\left(\frac{N^{\omega_{1}/3}}{N^{\omega_{% 0}/6}}\right)\varleq\frac{N^{4\varepsilon}}{N}\mathcal{O}\left(\frac{N^{\omega% _{1}/3}}{N^{\omega_{0}/6}}+\frac{1}{N^{\omega_{1}/60}}\right) (3.212)

In the case \omega_{1}<10\omega_{0}/21 we take \omega_{\ell}=21\omega_{1}/20<\omega_{0}/2. The error simplifies to

 \frac{N^{2\varepsilon}}{N}\mathcal{O}\left(\frac{1}{N^{\omega_{1}/20}}+\frac{1% }{N^{\varepsilon_{a}}}+N^{\varepsilon_{2}+\varepsilon_{a}}\frac{1}{N^{\omega_{% 1}/10}}+N^{\varepsilon_{a}-\varepsilon_{2}/2}\right) (3.213)

Choose \varepsilon_{2}=\omega_{1}/15 and \varepsilon_{a}=\omega_{1}/60. The error then simplifies to

 \frac{N^{3\varepsilon}}{N}\mathcal{O}\left(\frac{1}{N^{\omega_{1}/60}}\right)% \varleq\frac{N^{4\varepsilon}}{N}\mathcal{O}\left(\frac{1}{N^{\omega_{1}/60}}+% \frac{N^{\omega_{1}/3}}{N^{\omega_{0}/6}}\right). (3.214)

## 4 Finite speed estimates

### 4.1 Estimate for short-range operator

In this section we work in the set-up of Section 3.5 and assume that 1-3 hold. Let \hat{z}_{i} be defined as in that section. Fix a parameter \ell_{3}=N^{\omega_{\ell,3}} satisfying

 0<\omega_{\ell,3}\varleq\omega_{\ell}. (4.1)

For the current Section 4.1 we fix a 0<q<1 and let \mathcal{A}_{3} be the set

 \displaystyle\mathcal{A}_{3}:=\{(i,j):|i-j|\varleq N^{\omega_{\ell,3}}\}\cup\{% (i,j):ij>0,i\notin\hat{\mathcal{C}}_{q},j\notin\hat{\mathcal{C}}_{q}\}. (4.2)

Define the operator

 (\mathcal{B}_{3}u)_{i}:=\sum^{\mathcal{A}_{3},(i)}_{j}\frac{1}{N}\frac{u_{j}-u% _{i}}{(\hat{z}_{j}-\hat{z}_{i})^{2}}. (4.3)

We want to prove the following theorem. Lemma 3.9 is an immediate consequence. The method is based on that appearing in [18].

###### Theorem 4.1.

Let \ell_{3} as above. Let D_{1},D_{2}>0, and let \varepsilon>0. Let 0<q_{3}<q. We assume that Section 3.5 1-3 hold. Fix a time u\varleq 10t_{1}. There is an event \mathcal{F}(\alpha,u) s.t. \mathbb{P}[\mathcal{F}(\alpha,u)]\vargeq 1-N^{-D_{1}} for N large enough (independent of \alpha) such that all of the following estimates hold. For every u\varleq s\varleq t\varleq\left(u+2\ell_{3}/N\right)\wedge 10t_{1} we have the estimate

 \mathcal{U}^{\mathcal{B}_{3}}_{ba}(s,t)\varleq\frac{1}{N^{D_{2}}}. (4.4)

provided one of the following three criteria holds.

1. [label=()]

2. a\in\hat{\mathcal{C}}_{q_{3}} and |a-b|>N^{\omega_{\ell_{3}}+\varepsilon}.

3. b\in\hat{\mathcal{C}}_{q_{3}} and |a-b|>N^{\omega_{\ell,3}+\varepsilon}.

4. a\notin\hat{\mathcal{C}}_{q_{3}}, b\notin\hat{\mathcal{C}}_{q_{3}} and ab<0.

Hence, the same estimate holds for any 0\varleq s\varleq t\varleq 10t_{1} that satisfy t-s\varleq\ell_{3}/N, with overwhelming probability.

In the proof of Theorem 4.1 we take u=0 for notational simplicity. The first step in proving Theorem 4.1 is to establish the estimate for s=0, which is the content of the following lemma. We then use the semigroup property to extend the estimate to all s.

###### Lemma 4.2.

Fix 0<q_{3}<q. Let \varepsilon>0 and D_{1},D_{2}>0. Assume that Section 3.5 1-3 hold. There is an event \mathcal{F}_{\alpha} with \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D_{1}} on which the following estimates hold. For every 0\varleq t\varleq\ell_{3}/N\wedge 10t_{1} we have the estimate

 \mathcal{U}^{\mathcal{B}_{3}}_{ba}(0,t)\varleq\frac{1}{N^{D_{2}}}. (4.5)

provided one of the following three criteria holds.

1. [label=()]

2. a\in\hat{\mathcal{C}}_{q_{3}} and |a-b|>N^{\omega_{\ell_{3}}+\varepsilon}.

3. b\in\hat{\mathcal{C}}_{q_{3}} and |a-b|>N^{\omega_{\ell,3}+\varepsilon}.

4. a\notin\hat{\mathcal{C}}_{q_{3}}, b\notin\hat{\mathcal{C}}_{q_{3}} and ab<0.

Proof.  Define for t\vargeq 0, f_{i}(t):=\mathcal{U}^{B}_{ia}(0,t); i.e., f_{i} satisfies the equation

WLOG, take \varepsilon>0 s.t. \omega_{\ell,3}+\varepsilon<\omega_{0}/2. We can assume a\vargeq 0. Fix q_{4} satisfying q_{3}<q_{4}<q. It then suffices to prove the statement for the following two cases.

1. a\in\hat{\mathcal{C}}_{q_{4}}, |a-b|>N^{\omega_{\ell,3}+\varepsilon}.

2. a\notin\hat{\mathcal{C}}_{q_{4}}, b\in\hat{\mathcal{C}}_{q_{3}} or b<0.

Let us first consider the case a\in\hat{\mathcal{C}}_{q_{4}}. Let \nu>0 and define

 \phi_{k}:=\mathrm{e}^{\nu\psi(\hat{z}_{k}(t,\alpha)-\tilde{\gamma}_{a}(t,% \alpha))/2} (4.7)

where \psi is the following smooth function. Fix a scale \ell_{4}=N^{\omega_{\ell,4}}>0 and a \delta_{1}>0 s.t.

Assume 0<\varepsilon<\delta_{1}. We choose \psi s.t.

and

We can choose \psi so that |\psi^{\prime}|\varleq 1 and |\psi^{\prime\prime}|\varleq CN/\ell_{4}.

Our proof is based around a Gronwall argument and we will need to take an expectation of a martingale. For this we need to introduce the following stopping time \tau_{r}. Let q<q_{r}<1 and \varepsilon_{r}>0 with \varepsilon_{r}<\varepsilon/100. Let \tau_{i}, i=1,2,3 be the stopping time

 \displaystyle\tau_{1} \displaystyle:=\inf\{t>0:\exists i\in\hat{\mathcal{C}}_{q_{r}}:|\hat{z}_{i}(t)% -\gamma_{i}(t)|>N^{\varepsilon_{r}}/N\} \displaystyle\tau_{2} \displaystyle:=\inf\{t>0:\exists i\in\hat{\mathcal{C}}_{q_{r}}:|J_{i}|>C_{J}% \log(N)\} \displaystyle\tau_{3} \displaystyle:=\inf\{t>0:\exists i:|F_{i}|\vargeq\log(N)\}. (4.11)

We set \tau_{r}:=\tau_{1}\wedge\tau_{2}\wedge\tau_{3}\wedge(10t_{1}). We know that \tau_{r}=10t_{1} with overwhelming probability by the assumptions 2 and 3 of Section 3.5.

Define now v_{k}(t)=\phi_{k}f_{k} and F=\sum_{k}v^{2}_{k}(t)\boldsymbol{1}_{\{\tau_{r}>0\}}. By the same calculation as in [18] we obtain

 \displaystyle\mathrm{d}F(t) \displaystyle=-\frac{1}{2}\sum_{(i,j)\in\mathcal{A}_{3}}\frac{1}{N}\frac{(v_{k% }-v_{j})^{2}}{(\hat{z}_{k}-\hat{z}_{j})^{2}}\mathrm{d}t (4.12) \displaystyle-\sum_{(j,k)\in\mathcal{A}_{3}}\frac{1}{N}\frac{1}{(\hat{z}_{j}-% \hat{z}_{k})^{2}}\left[\frac{\phi_{k}}{\phi_{j}}+\frac{\phi_{j}}{\phi_{k}}-2% \right]v_{k}v_{j}\mathrm{d}t (4.13) \displaystyle+\sum_{k}\nu v_{k}^{2}\psi_{k}^{\prime}\mathrm{d}(\hat{z}_{k}-% \tilde{\gamma}_{a}) (4.14) \displaystyle+\sum_{k}v_{k}^{2}\left(\nu^{2}(\psi_{k}^{\prime})^{2}+\nu\psi^{% \prime\prime}_{k}\right)\frac{\mathrm{d}t}{N}. (4.15)

We now deal with each term individually, applying Gronwall at the end of the proof. In the remainder of the argument we work on times t<\tau_{r}. We start with (4.13). Fix q_{5} satisfying q_{4}<q_{5}<q. By rigidity and choice of \psi we have that the term

vanishes unless j,k\in\hat{\mathcal{C}}_{q_{5}}. In this case by rigidity and the fact that |\psi^{\prime}|\varleq 1 we have that

 \left|\frac{\phi_{k}}{\phi_{j}}+\frac{\phi_{j}}{\phi_{k}}-2\right|\varleq C\nu% ^{2}|\hat{z}_{j}-\hat{z}_{k}|^{2} (4.17)

as long as we choose \nu so that \nu\ell_{3}\varleq CN. Hence,

 \left|\sum_{(j,k)\in\mathcal{A}_{3}}\frac{1}{N}\frac{1}{(\hat{z}_{j}-\hat{z}_{% k})^{2}}\left[\frac{\phi_{k}}{\phi_{j}}+\frac{\phi_{j}}{\phi_{k}}-2\right]v_{k% }v_{j}\right|\varleq C\frac{\nu^{2}}{N}\ell_{3}\sum_{k}v_{k}^{2}. (4.18)

Above we used the fact that the cardinality of the set \{j:(j,k)\in\mathcal{A}_{3}\} is bounded by C\ell_{3} if k\in\hat{\mathcal{C}}_{q_{5}}. The Ito terms (4.15) are bounded by

 \left|\sum_{k}v_{k}^{2}\left(\nu^{2}(\psi_{k}^{\prime})^{2}+\nu\psi^{\prime% \prime}_{k}\right)\frac{\mathrm{d}t}{N}\right|\varleq C\left(\frac{\nu^{2}}{N}% +\frac{\nu}{\ell_{4}}\right)\sum_{k}v_{k}^{2}. (4.19)

We now deal with the terms (4.14). By rigidity we have \psi^{\prime}_{k}=0 if k\notin\hat{\mathcal{C}}_{q_{5}}. We can therefore assume k\in\hat{\mathcal{C}}_{q_{5}}. We fix a \delta_{2}>0 s.t.

 \delta_{2}<\omega_{\ell,3}. (4.20)

From the definition of the \hat{z}_{k} process and the definition of \tau_{r} we see that we can write for k\in\hat{\mathcal{C}}_{q_{5}},

 \mathrm{d}(\hat{z}_{k}-\tilde{\gamma}_{a})=\sum_{|j-k|\varleq N^{\delta_{2}}}% \frac{1}{N}\frac{1}{\hat{z}_{k}-\hat{z}_{j}}\mathrm{d}t+X_{t}\mathrm{d}t+\sqrt% {\frac{2}{N}}\mathrm{d}B_{k} (4.21)

where we have the bound |X_{t}|\varleq C\log N. The first term on the RHS of (4.21) corresponds to

 \displaystyle\nu\sum_{k}v_{k}^{2}\psi^{\prime}_{k}\sum_{|j-k|\varleq N^{\delta% _{2}}}\frac{1}{N}\frac{1}{\hat{z}_{k}-\hat{z}_{j}} \displaystyle=\frac{\nu}{2}\sum_{|j-k|\varleq N^{\delta_{2}}}\frac{\psi^{% \prime}_{k}}{N}\frac{v_{k}^{2}-v_{j}^{2}}{\hat{z}_{k}-\hat{z}_{j}}+\frac{\nu}{% 2}\sum_{|j-k|\varleq N^{\delta_{2}}}\frac{v_{k}^{2}}{N}\frac{\psi^{\prime}_{j}% -\psi^{\prime}_{k}}{\hat{z}_{k}-\hat{z}_{j}} (4.22)

The second term of (4.22) is bounded by

 \left|\frac{\nu}{2}\sum_{|j-k|\varleq N^{\delta_{2}}}\frac{v_{k}^{2}}{N}\frac{% \psi^{\prime}_{j}-\psi^{\prime}_{k}}{\hat{z}_{k}-\hat{z}_{j}}\right|\varleq C% \frac{\nu N^{\delta_{2}}}{\ell_{4}}\sum_{k}v_{k}^{2}. (4.23)

We use the Schwarz inequality to bound the first term of (4.22) by

 \left|\frac{\nu}{2}\sum_{|j-k|\varleq N^{\delta_{2}}}\frac{\psi^{\prime}_{k}}{% N}\frac{v_{k}^{2}-v_{j}^{2}}{\hat{z}_{k}-\hat{z}_{j}}\right|\varleq\frac{1}{10% 0}\sum_{|j-k|\varleq N^{\delta_{2}}}\frac{(v_{k}-v_{j})^{2}}{N(\hat{z}_{k}-% \hat{z}_{j})^{2}}+\frac{C\nu^{2}}{N}N^{\delta_{2}}\sum_{k}v_{k}^{2}. (4.24)

The first term on the RHS is absorbed into the term (4.12). Collecting everything we have proven that under the assumption \nu\ell_{3}\varleq CN, we have

 \partial_{t}\mathbb{E}[F(t)]\varleq C\left(\frac{\nu^{2}\ell_{3}}{N}+\frac{\nu N% ^{\delta_{2}}}{\ell_{4}}+\nu\log(N)\right)\mathbb{E}[F(t)]. (4.25)

We can take \ell_{4}=\ell_{3} and \nu=N/(\ell_{3}N^{\varepsilon/2}). Then by Gronwall we see that for t\varleq\ell_{3}/N we get

 \mathbb{E}[F(t)]\varleq C\mathbb{E}[F(0)]\varleq C (4.26)

where the second inequality follows from rigidity, the definition of \tau and the initial condition f_{k}(0)=\delta_{ak}. By construction, for any \varepsilon>0 we have that if j\varleq a-N^{\omega_{\ell,3}+\varepsilon},

 \psi(\hat{z}_{j}-\tilde{\gamma}_{a})>\frac{c}{N}\min\{N^{\omega_{\ell,3}+% \varepsilon},N^{\omega_{\ell,3}+\delta_{1}}\}=\frac{c}{N}N^{\omega_{\ell,3}+\varepsilon} (4.27)

and hence

 \nu\psi(\hat{z}_{j}-\tilde{\gamma}_{a})>cN^{\varepsilon/2}. (4.28)

We conclude the claim for b\varleq a-N^{\omega_{\ell,3}-\varepsilon} and a\in\hat{\mathcal{C}}_{q_{4}} from Markov’s inequality. For b>a+N^{\omega_{\ell}+\varepsilon} the argument is similar; one just replaces \psi by -\psi.

We now consider the case a\notin\hat{\mathcal{C}}_{q_{4}}. Recall that we assumed a\vargeq 0. The argument is identical except one considers, instead of \psi above,

 \varphi(x):=\psi(x-\tilde{\gamma}_{d}(t,\alpha)) (4.29)

where d is an index chosen in the following way. If d_{1}>0 is the largest index in \hat{\mathcal{C}}_{q_{3}} and d_{2}>0 is the largest index in \hat{\mathcal{C}}_{q_{4}} (recall q_{3}<q_{4}) then d=(d_{1}+d_{2})/2. One then defines

With this choice rigidity implies that \varphi^{\prime}_{k}=0 for k\notin\hat{\mathcal{C}}_{q_{4}}. Rigidity also implies that the term

vanishes unless both j,k\in\hat{\mathcal{C}}_{q_{4}}. With these considerations the argument can proceed exactly as above. We again arrive at (4.25) and choose \ell_{4}=\ell_{3} and \nu=N/(\ell_{3}N^{\varepsilon/2}). We see that for k\in\hat{\mathcal{C}}_{q_{3}} we have

 \psi(\hat{z}_{k}-\tilde{\gamma}_{d})\vargeq cN^{\delta_{1}+\omega_{\ell}}/N (4.32)

and so \nu\psi(\hat{z}_{k}-\tilde{\gamma}_{d})\vargeq cN^{\varepsilon/2}. We conclude as before. Note that now we only need that \psi(\hat{z}_{a}-\tilde{\gamma}_{d})<0 which follows by the ordering of particles and rigidity to satisfy

 \mathbb{E}[F(0)]\varleq C. (4.33)

Proof of Theorem 4.1. For notational simplicity we set u=0. Let \varepsilon and q_{3} be as in the statement of Theorem 4.1. Wlog, we can assume that \omega_{\ell,3}+\varepsilon<\omega_{0}/2. We can assume that the estimates of Lemma 4.2 hold for q_{4} satisfying q_{3}<q_{4}<q and \varepsilon^{\prime}=\varepsilon/2. For any i we can write

 \mathcal{U}^{\mathcal{B}_{3}}_{bi}(0,t)=\sum_{j}\mathcal{U}^{\mathcal{B}_{3}}_% {bj}(s,t)\mathcal{U}^{\mathcal{B}_{3}}_{ji}(0,s)\vargeq\mathcal{U}^{\mathcal{B% }_{3}}_{ba}(s,t)\mathcal{U}^{\mathcal{B}_{3}}_{ai}(0,s). (4.34)

We just need to find an i s.t. the LHS is bounded above and \mathcal{U}^{\mathcal{B}_{3}}_{ai}(0,s) is bounded below. Fix q_{5} satisfying q_{3}<q_{5}<q_{4}. As before, it suffices to assume a\vargeq 0 and to consider the following two cases.

1. a\in\hat{\mathcal{C}}_{q_{5}}, and |b-a|>N^{\omega_{\ell,3}+\varepsilon}.

2. a\notin\hat{\mathcal{C}}_{q_{5}} and b\in\hat{\mathcal{C}}_{q_{3}} or b<0.

Let us first consider the case a\in\hat{\mathcal{C}}_{q_{5}}. Since the estimates of Lemma 4.2 hold, and a\in\hat{\mathcal{C}}_{q_{5}}, we have that

Since

 \sum_{i}\mathcal{U}^{\mathcal{B}_{3}}_{ai}(0,s)=1, (4.36)

this implies that there is an i_{0} s.t. |i_{0}-a|\varleq N^{\omega_{\ell,3}+\varepsilon/2} and \mathcal{U}^{B}_{ai_{0}}(0,s)\vargeq N^{-1}. Moreover, i_{0}\in\hat{\mathcal{C}}_{q_{4}}. Then since |a-b|>N^{\omega_{\ell,3}+\varepsilon} we see that |i_{0}-b|>N^{\omega_{\ell,3}+\varepsilon/2}. Hence,

 \mathcal{U}^{\mathcal{B}_{3}}_{bi_{0}}(0,t)\varleq\frac{1}{N^{D_{2}}}. (4.37)

Therefore,

 \mathcal{U}^{\mathcal{B}_{3}}_{ba}(s,t)\varleq\frac{1}{N^{D_{2}-1}}. (4.38)

Let us now consider the case a\notin\hat{\mathcal{C}}_{q_{5}} and b\in\hat{\mathcal{C}}_{q_{3}} or b<0. Wlog we can take a>0. Fix q_{6} s.t. q_{3}<q_{6}<q_{5}. If i is such that either i\in\hat{\mathcal{C}}_{q_{6}} or i\varleq 0, then \mathcal{U}^{\mathcal{B}_{3}}_{ai}(0,s)\varleq N^{-D_{2}}. Hence, there is an i_{0} s.t. \mathcal{U}^{\mathcal{B}_{3}}_{ai_{0}}\vargeq N^{-1} and i_{0}>0 and i_{0}\notin\hat{\mathcal{C}}_{q_{6}}. But then since b\in\hat{\mathcal{C}}_{q_{3}} or b<0 we get that

 \mathcal{U}^{\mathcal{B}_{3}}_{bi_{0}}(0,t)\varleq\frac{1}{N^{D_{2}}} (4.39)

and this yields the claim as before. ∎

### 4.2 Kernel estimate

In this section we prove Lemma 3.10. It is split into two parts, an energy estimate and a Duhamel expansion.

#### 4.2.1 Energy estimate

Let \mathcal{B} be as in Section 3.3. In this subsection our goal is to prove the following energy estimate for \mathcal{U}^{B}. The argument is very similar to that in [40]. The major difference is that in the duality part of Nash’s argument we have to be a little careful with the support of the functions, as we do not know that rigidity holds for all particles i. To compensate for this we use the finite speed estimates from the previous section.

We recall the semigroup \mathcal{U}^{B} for the short-range operator \mathcal{B} associated with in short-range set \mathcal{A}_{q_{*}} with parameters q_{*} and \omega_{\ell} from Section 3.5.

###### Lemma 4.3.

Fix 0<q_{3}<q_{*}. Let a\in\hat{\mathcal{C}}_{q_{3}}. Let \varepsilon>0,D>0. Assume that Section 3.5 1-3 hold. There is an event \mathcal{F}_{\alpha} which holds with probability \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimates hold. For every 0\varleq s\varleq t\varleq 10t_{1} and every i,

 \mathcal{U}^{B}_{ia}(s,t)\varleq\frac{N^{\varepsilon}}{N(t-s)}. (4.40)

Proof.  Recall from [40] the inequality

 ||u||_{4}^{4}||u||_{2}^{-2}\varleq C\sum_{i,j\in\mathbb{Z}}\frac{(u_{i}-u_{j})% ^{2}}{(i-j)^{2}} (4.41)

which holds for sequences u:\mathbb{Z}\to\mathbb{R}. Fix q_{3}<q_{5}<q_{*}. Let g(u)=\mathcal{U}^{B}(s,u)g(s) where g(s) has support only in the indices i\in\hat{\mathcal{C}}_{q_{5}} and

 ||g(s)||_{1}=1. (4.42)

Extending g(u) by 0 to all of \mathbb{Z} we apply the above inequality to g(u) and obtain, with overwhelming probability,

 \displaystyle||g(u)||_{4}^{4}||g(u)||_{2}^{-2} \displaystyle\varleq C\sum_{i,j\in\mathbb{Z}}\frac{(g_{i}(u)-g_{j}(u))^{2}}{(i% -j)^{2}}\varleq C\sum_{|i-j|\varleq\ell,i,j\in\hat{\mathcal{C}}_{q_{*}}}\frac{% (g_{i}(u)-g_{j}(u))^{2}}{(i-j)^{2}}+C\frac{N}{\ell}||g(u)||_{2}^{2}+\frac{1}{N% ^{D}} \displaystyle\varleq N^{\varepsilon}\langle g(u),\mathcal{B}g(u)\rangle+C||g(u% )||_{2}^{2}\frac{N}{\ell}+\frac{1}{N^{D}}. (4.43)

We used the fact that for 0\varleq u_{1}\varleq u_{2}\varleq t_{1},

which holds due to Theorem 4.1, with overwhelming probability. Therefore

 \partial_{t}||g(u)||_{2}^{2}=-\langle g(u),\mathcal{B}g(u)\rangle\varleq-N^{-% \varepsilon}||g(u)||_{2}^{4}+C||g(u)||_{2}^{2}\frac{N}{\ell}+\frac{1}{N^{D}}. (4.45)

Above, we used the Holder inequality ||g(u)||_{2}\varleq||g(u)||_{1}^{1/3}||g(u)||_{4}^{2/3}\varleq||g(s)||_{1}^{1/% 3}||g(u)||_{4}^{2/3}=||g(u)||_{4}^{2/3} for u\vargeq s. Since t_{1}N/\ell\ll 1 we see that this implies

 ||g(t)||_{2}\varleq CN^{\varepsilon}(t-s)^{-1/2}. (4.46)

We have therefore proven that

 ||\mathcal{U}^{B}(s,t)g||_{2}\varleq(t-s)^{-1/2}N^{\varepsilon}||g||_{1} (4.47)

for every g supported in i\in\hat{\mathcal{C}}_{q_{5}}.

The above argument clearly also applies to (\mathcal{U}^{B}(s,t))^{T} (in particular note that the bound (4.44) is symmetric in i and j).

Fix now q_{4} satisfying q_{3}<q_{4}<q_{5}. Let now f be supported in i\in\hat{\mathcal{C}}_{q_{4}} and have ||f||_{2}=1. Then we have with overwhelming probability,

 ||\mathcal{U}^{B}(s,t)f||_{\infty}=\sup_{||g||_{1}=1}\langle\mathcal{U}^{B}(s,% t)f,g\rangle\varleq\sup_{||g||_{1}=1,g_{i}=0,i\notin\hat{\mathcal{C}}_{q_{5}}}% \langle\mathcal{U}^{B}(s,t)f,g\rangle+\frac{1}{N^{D}} (4.48)

where we used the fact that |(\mathcal{U}^{B}(s,t)f)_{i}|\varleq N^{-D} for any D>0 for i\notin\hat{\mathcal{C}}_{q_{5}} which holds due to Theorem 4.1. For g as in the RHS of (4.48) we have,

 \langle\mathcal{U}^{B}(s,t)f,g\rangle=\langle f,(\mathcal{U}^{B}(s,t))^{T}g% \rangle\varleq||f||_{2}||(\mathcal{U}^{B}(s,t))^{T}g||_{2}\varleq N^{% \varepsilon}(t-s)^{-1/2}. (4.49)

This proves that for f supported in i\in\hat{\mathcal{C}}_{q_{4}} we have

 ||\mathcal{U}^{B}(s,t)f||_{\infty}\varleq N^{\varepsilon}(t-s)^{-1/2}||f||_{2}. (4.50)

Lastly, let f have support in \hat{\mathcal{C}}_{q_{3}} and ||f||_{1}=1. Applying what we have proved above we have with overwhelming probability, with u=s+(t-s)/2,

 \displaystyle||\mathcal{U}^{B}(s,t)f||_{\infty} \displaystyle=||\mathcal{U}^{B}(u,t)\mathcal{U}^{B}(s,u)f||_{\infty}\varleq||% \mathcal{U}^{B}(u,t)\boldsymbol{1}_{\hat{\mathcal{C}}_{q_{4}}}\mathcal{U}^{B}(% s,u)f||_{\infty}+\frac{1}{N^{D}} \displaystyle\varleq C\frac{N^{\varepsilon}}{(t-s)^{1/2}}||\boldsymbol{1}_{% \hat{\mathcal{C}}_{q_{4}}}\mathcal{U}^{B}(s,u)f||_{2}+\frac{1}{N^{D}}\varleq C% \frac{N^{\varepsilon}}{(t-s)^{1/2}}||\mathcal{U}^{B}(s,u)f||_{2}+\frac{1}{N^{D}} \displaystyle\varleq C\frac{N^{2\varepsilon}}{t-s}||f||_{1}. (4.51)

This yields the claim. ∎

#### 4.2.2 Duhamel expansion

We want to prove the following.

###### Lemma 4.4.

Let D>0 and \varepsilon_{1},\varepsilon_{2},\varepsilon_{3}>0 and 0<q_{3}<q_{*}. Assume that Section 3.5 1 - 3 hold. Let \ell_{3} be a scale satisfying

 N^{\varepsilon_{3}}\varleq\ell_{3}\varleq N. (4.52)

There exists an event \mathcal{F}_{\alpha}=\mathcal{F}_{\alpha}(\ell_{3}) with probability \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimates hold. For every 0\varleq s\varleq t\varleq 10t_{1} which satisfy N(t-s)\varleq N^{-\varepsilon_{1}}\ell_{3} and indices a and p satisfying a,p\in\hat{\mathcal{C}}_{q_{3}} and

 |a-p|\vargeq N^{\varepsilon_{1}}\ell_{3} (4.53)

we have

 \mathcal{U}^{B}_{ap}(s,t)\varleq N^{\varepsilon_{2}}\frac{N(t-s)+1}{(a-p)^{2}}. (4.54)

By taking a sequence of at most N scales \ell_{3}=N^{\varepsilon},N^{\varepsilon}+1,\cdots we easily see that Lemma 4.4 implies the following estimate.

###### Lemma 4.5.

Let D>0 and \varepsilon_{1},\varepsilon_{2} and 0<q_{3}<q_{*}. There is an event \mathcal{F}_{\alpha} with probability \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following holds. For every 0\varleq s\varleq t\varleq 10t_{1} and pair of indices a,p satisfying a,p\in\hat{\mathcal{C}}_{q_{3}} and

 |a-p|\vargeq N^{\varepsilon_{1}}\left[1\vee(N(t-s))\right] (4.55)

we have

 \mathcal{U}^{B}_{ap}(s,t)\varleq N^{\varepsilon_{2}}\frac{N(t-s)+1}{(a-p)^{2}}. (4.56)

Proof of Lemma 4.4. We can work under the assumption that the estimates of Lemma 3.5 hold. We assume a<p. The proof for a>p is identical. Write

 \mathcal{B}=\mathcal{S}+\mathcal{R} (4.57)

where \mathcal{S}=\mathcal{B}_{3} is defined as at the start of Section 4.1 with \ell_{3} as in the statement of Lemma 4.4, and \mathcal{R} is defined implicitly by the above equality. For notational simplicity we set s=0, but this has no effect on the proof. For each M we have

 \displaystyle\mathcal{U}^{B}(0,t) \displaystyle=\mathcal{U}^{\mathcal{S}}(0,t)+\sum_{i=1}^{M}\int_{0\varleq s_{1% }\varleq\cdots s_{i}\varleq t}\mathcal{U}^{\mathcal{S}}(s_{i},t)\mathcal{R}% \mathcal{U}^{\mathcal{S}}(s_{i-1},s_{i})\cdots\mathcal{R}\mathcal{U}^{\mathcal% {S}}(0,s_{1})\mathrm{d}s_{1}\cdots\mathrm{d}s_{i} \displaystyle+\int_{0\varleq s_{1}\cdots\varleq s_{M+1}\varleq t}\mathcal{U}^{% B}(s_{M+1},t)\mathcal{R}\mathcal{U}^{\mathcal{S}}(s_{M},s_{M+1})\cdots\mathcal% {R}\mathcal{U}^{\mathcal{S}}(0,s_{1})\mathrm{d}s_{1}\cdots\mathrm{d}s_{M+1} \displaystyle=:\mathcal{U}^{\mathcal{S}}(0,t)+\sum_{i=1}^{M}\int_{0\varleq s_{% 1}\cdots\varleq t}A_{i}\mathrm{d}s_{1}\cdots\mathrm{d}s_{i}+B_{M+1}. (4.58)

By Theorem 4.1 we have \mathcal{U}^{\mathcal{S}}_{ap}(0,t)\varleq N^{-100}. We next deal with the term B_{M+1}. Using the estimates of Lemma 3.5, it is easy to check that for every i we have

 \sum_{j}|\mathcal{R}(i,j)|\varleq C\frac{N}{\ell_{3}} (4.59)

and so

 ||\mathcal{R}||_{\ell^{p}\to\ell^{p}}\varleq C\frac{N}{\ell_{3}} (4.60)

for every p. Hence,

 |B_{M+1}(i,j)|\varleq C_{M}N^{2}\left(\frac{Nt}{\ell_{3}}\right)^{M}\varleq% \frac{1}{N^{100}} (4.61)

for M a large constant depending only on \varepsilon_{1}. By Lemma 4.6 below we see that

 \left|\int_{0\varleq s_{1}\cdots\varleq t}A_{i}(a,p)\mathrm{d}s_{1}\cdots% \mathrm{d}s_{i}\right|\varleq C\frac{Nt}{(a-p)^{2}}\left(\frac{Nt}{\ell_{3}}% \right)^{i-1} (4.62)

Note that we used that i\varleq M and so a\varleq L_{i}. To estimate the integral we used the bound (4.64) on the integrand and that the region of integration has size Ct^{i}. This concludes the proof. ∎

###### Lemma 4.6.

Let A_{i}, \mathcal{R}, etc. be as above. Let a<p, M be as above. Define L_{i} as

 L_{i}:=p-i\frac{p-a}{100M}. (4.63)

Then for j\varleq L_{i} we have

 A_{i}(j,p)\varleq C_{M}\frac{N^{i}}{(p-a)^{2}\ell_{3}^{i-1}}. (4.64)

Remark. The proof will be by induction on i, using the estimate from the case i-1. We rewrite A_{i+1}(j,p) in terms of a sum which involes matrix elements A_{i}(l,p). If j\varleq L_{i+1} and l\vargeq L_{i} then by definition |l-j|\vargeq c|p-a| and so we can apply Theorem 4.1 to simplify the sum. In the case that l\varleq L_{i} we can use the induction assumption. Proof.  The proof is by induction on i. Define

 R:=N^{\varepsilon_{1}/4}\ell_{3}. (4.65)

Fix q_{3}<q_{5}<q_{*}. Theorem 4.1 implies that for (a,b) s.t. |a-b|>R and either a or b\in\hat{\mathcal{C}}_{q_{5}}, we have with overwhelming probability

 |\mathcal{U}^{\mathcal{S}}_{ab}(s,t)|\varleq\frac{1}{N^{D}} (4.66)

for any D>0, and 0\varleq s\varleq t\varleq\ell_{3}. We have that

 A_{1}(j,p)=\sum_{k,l}\mathcal{U}^{\mathcal{S}}_{jk}\mathcal{R}_{kl}\mathcal{U}% ^{\mathcal{S}}_{lp}. (4.67)

Since the matrix elements of \mathcal{R} are bounded by (say) N^{2}, it suffices by (4.66) to consider only terms in (4.67) that satisfy |l-p|\varleq R. Since j\varleq L_{1} we can apply Theorem 4.1 and ignore terms satisfying

 k\vargeq L_{1}+R. (4.68)

For such l and k, using the fact that j\varleq L_{1} and R\ll|p-a| we see that |l-k|\vargeq c(p-a) and so

 |A_{1}(j,p)|\varleq\sum_{k,l}\mathcal{U}^{\mathcal{S}}_{j,k}\frac{N}{(p-a)^{2}% }\mathcal{U}^{\mathcal{S}}_{l,p}\varleq\frac{N}{(p-a)^{2}}, (4.69)

where in the second inequality we used the \ell^{p}\to\ell^{p} boundedness of \mathcal{U}^{\mathcal{S}}. For \mathcal{R} we used in the first inequality that for |l-k|\vargeq c(p-a),

 \mathcal{R}(k,l)=\frac{1}{N}\frac{1}{(\hat{z}_{k}-\hat{z}_{l})^{2}}\varleq% \frac{CN}{(p-a)^{2}} (4.70)

with overwhelming probability. Now for A_{i+1} and j\varleq L_{i+1} we write

 A_{i+1}(j,p)=\sum_{k}\sum_{l\vargeq L_{i}}\mathcal{U}^{\mathcal{S}}(j,k)% \mathcal{R}(k,l)A_{i}(l,p)+\sum_{k}\sum_{l\varleq L_{i}}\mathcal{U}^{\mathcal{% S}}(j,k)\mathcal{R}(k,l)A_{i}(l,p) (4.71)

We start with estimating the first sum. By Theorem 4.1 we can restrict the k summation to terms satisfying

 k\varleq L_{i+1}+R. (4.72)

Then since l\vargeq L_{i} we get that |k-l|\vargeq c|p-a| and so

 \displaystyle\left|\sum_{k}\sum_{l\vargeq L_{i}}\mathcal{U}^{\mathcal{S}}(j,k)% \mathcal{R}(k,l)A_{i}(l,p)\right| \displaystyle\varleq C\frac{N}{(p-a)^{2}}\sum_{k,l}\mathcal{U}^{\mathcal{S}}(j% ,k)A_{i}(l,p)\varleq C\frac{N}{(p-a)^{2}}||\mathcal{U}^{\mathcal{S}}||_{\infty% \to\infty}||A_{i-1}||_{1\to 1} \displaystyle\varleq C\frac{N}{(p-a)^{2}}\frac{N^{i}}{\ell_{3}^{i}}. (4.73)

For the second sum in (4.71) we apply the induction assumption and obtain

 \displaystyle\left|\sum_{k}\sum_{l\varleq L_{i}}\mathcal{U}^{\mathcal{S}}(j,k)% \mathcal{R}(k,l)A_{i}(l,p)\right| \displaystyle\varleq\sum_{k}\sum_{l\varleq L_{i}}\mathcal{U}^{\mathcal{S}}(j,k% )\mathcal{R}(k,l)C\frac{N^{i}}{\ell^{i-1}(p-a)^{2}}\varleq C||\mathcal{U}^{% \mathcal{S}}\mathcal{R}||_{\infty\to\infty}\frac{N^{i}}{\ell_{3}^{i-1}(p-a)^{2}} \displaystyle\varleq C\frac{N}{\ell_{3}}\frac{N^{i}}{\ell_{3}^{i-1}(p-a)^{2}}. (4.74)

This completes the proof. ∎

### 4.3 Profile of random kernel

Combining Lemma 4.3 and 4.5 we easily obtain the following.

###### Theorem 4.7.

Fix 0<q_{3}<q_{*}. Let D and \varepsilon>0. Assume that Section 3.5 1-3 hold. There is an event \mathcal{F}_{\alpha} with probability \mathbb{P}[\mathcal{F}_{\alpha}]\vargeq 1-N^{-D} on which the following estimate holds. We have for every 0\varleq s\varleq t\varleq t_{1} and j,k\in\hat{\mathcal{C}}_{q_{3}},

 \mathcal{U}^{B}_{ij}(s,t)\varleq N^{\varepsilon}\frac{|t-s|\vee N^{-1}}{((i-j)% /N)^{2}+((t-s)\wedge N^{-1})^{2}} (4.75)

## 5 Regularity of hydrodynamic equation

In this section we analyze the limiting equation (3.88), deriving in particular the estimates in Lemma 3.12. We introduce the kernel

 K_{\eta_{1}}f(x):=\int_{|x-y|\leq\eta_{1}}\frac{f(x)-f(y)}{(x-y)^{2}}\mathrm{d% }y. (5.1)

The integral is understood in a principal value sense.

By [25, Theorem 1.4], for the heat kernel corresponding to K_{1}, we have the estimates:

for |x-y|<1 and

for |x-y|\geq 1.

If f(t,x) satisfies \partial_{t}f(t,x)=K_{1}f(t,x), then g(t,x)=f(t/\eta_{\ell},x/\eta_{\ell}) satisfies \partial_{t}g(t,x)=K_{\eta_{\ell}}g(t,x), so from (5.2), (5.3) we obtain:

when |x-y|<\eta_{1} and

 p_{\eta_{\ell}}(t,x,y)\asymp\frac{1}{\eta_{\ell}}e^{-c|x-y|/\eta_{\ell}\log% \frac{|x-y|}{t}}, (5.5)

for t\leq\eta_{\ell} and |x-y|\geq\eta_{\ell}. (3.94) and (3.95) follow directly from this. In the rest of the proof, we will also use that the upper bound in (5.4) remains true for all x,y, t\leq\eta_{\ell} (See [24, Proposition 2.2, i)].):

We will first estimate the spatial derivatives of p_{\eta_{\ell}}(t,x,y). Letting \Lambda=(-\Delta)^{1/2} be the half-Laplacian with kernel

 \int\frac{f(x)-f(y)}{(x-y)^{2}}\mathrm{d}y.

The corresponding heat kernel is

 p_{\Lambda}(t,x,y):=(e^{t\Lambda}\delta)(x)=\frac{1}{\pi}\frac{t}{t^{2}+(x-y)^% {2}}. (5.7)

We have the Duhamel formula

 \displaystyle p_{\eta_{\ell}}(t,x,y) \displaystyle=(e^{tK_{\eta_{\ell}}}\delta_{y})(x)=p_{\Lambda}(t,x,y)+\int_{0}^% {t}e^{(t-s)\Lambda}(K_{\eta_{\ell}}-\Lambda)f_{\eta_{\ell}}(s)\,\mathrm{d}s. (5.8)

Here we have denoted for simplicity

 f_{\eta_{\ell}}(x,s):=p_{\eta_{\ell}}(s,x,y).

Since e^{t\Lambda} is smoothing, the equality (5.8) shows in particular that p_{\eta}(t,\cdot,y) is in C^{\infty}(\mathbb{R}).

We will estimate the first spatial derivative by differentiating (5.8). The operators \Lambda, K_{\eta_{\ell}} are translation invariant, so for general k we have

 \partial^{k}_{x}\int_{0}^{t}e^{(t-s)\Lambda}(K_{\eta_{\ell}}-\Lambda)f_{\eta_{% \ell}}(s)\,\mathrm{d}s=\int_{0}^{t}e^{(t-s)\Lambda}\partial^{k}_{x}(K_{\eta_{% \ell}}-\Lambda)f_{\eta_{\ell}}(s)\,\mathrm{d}s.

By direct computation we have

 (K_{\eta_{\ell}}-\Lambda)f(x)=\frac{2}{\eta_{\ell}}f(x)-\int_{|x-z|\geq\eta_{% \ell}}\frac{f(z)}{(x-z)^{2}}\,\mathrm{d}z.

Differentiating, we find,

 \partial^{k}_{x}(K_{\eta_{\ell}}-\Lambda)f(x)=\frac{2}{\eta_{\ell}}f^{(k)}(x)+% (-1)^{k+1}(k!)\int_{|x-z|\geq\eta_{\ell}}{f(z)}{(x-z)^{-k-2}}\,\mathrm{d}z. (5.9)

We will first derive an estimate on the second term. In order to estimate the first we later derive a Hölder estimate for p_{\eta_{\ell}}(t,x,y).

We first derive the following estimate.

###### Lemma 5.1.

We have,

 \int_{|x-z|\vargeq\eta_{\ell}}\frac{|{f_{\eta_{\ell}}}(z)|}{|x-z|^{k+2}}% \mathrm{d}z\varleq\frac{C}{\eta_{\ell}^{k+1}}\left(\frac{s}{s^{2}+(x-y)^{2}}+% \frac{\eta_{\ell}}{\eta_{\ell}^{2}+(x-y)^{2}}\right). (5.10)

for 0\varleq s\varleq t\varleq\eta_{\ell}.

Proof.  By (5.6) we can estimate

 \displaystyle\int_{|x-z|\vargeq\eta_{\ell}}\frac{|{f_{\eta_{\ell}}}(z)|}{|x-z|% ^{k+2}}\mathrm{d}z| \displaystyle\varleq C\int_{|z|\vargeq\eta_{\ell},|z|\varleq\frac{1}{2}|x-y|}% \frac{1}{|z|^{k+2}}\frac{s}{s^{2}+(z-(x-y))^{2}}\mathrm{d}z \displaystyle+C\int_{|z|\vargeq\eta_{\ell},|z|\vargeq\frac{1}{2}|x-y|}\frac{1}% {|z|^{k+2}}\frac{s}{s^{2}+(z-(x-y))^{2}}\mathrm{d}z (5.11)

For the first term we have

 \int_{|z|\vargeq\eta_{\ell},|z|\varleq\frac{1}{2}|x-y|}\frac{1}{|z|^{k+2}}% \frac{s}{s^{2}+(z-(x-y))^{2}}\mathrm{d}z\varleq C\frac{s}{s^{2}+(x-y)^{2}}\int% _{|z|\vargeq\eta_{\ell}}\frac{1}{|z|^{k+2}}\varleq\frac{C}{\eta_{\ell}^{k+1}}% \frac{s}{s^{2}+(x-y)^{2}}. (5.12)

For the second term we first consider the case |x-y|\vargeq\eta_{\ell}/2. We have,

 \int_{|z|\vargeq\eta_{\ell},|z|\vargeq\frac{1}{2}|x-y|}\frac{1}{|z|^{k+2}}% \frac{s}{s^{2}+(z-(x-y))^{2}}\mathrm{d}z\varleq\frac{C}{|x-y|^{k+2}}\int\frac{% s}{s^{2}+z^{2}}\mathrm{d}z\varleq\frac{C}{|x-y|^{k+2}}\varleq\frac{C}{\eta_{% \ell}^{k+1}}\frac{\eta_{\ell}}{\eta_{\ell}^{2}+(x-y)^{2}}. (5.13)

In the case |x-y|\varleq\eta_{\ell}/2 we have |z-(x-y)|\vargeq c|z| for |z|\vargeq\eta_{\ell} and so

 \int_{|z|\vargeq\eta_{\ell},|z|\vargeq\frac{1}{2}|x-y|}\frac{1}{|z|^{k+2}}% \frac{s}{s^{2}+(z-(x-y))^{2}}\mathrm{d}z\varleq\frac{Cs}{\eta_{\ell}^{k+1}}% \int_{|z|\vargeq\eta_{\ell}}\frac{1}{|z|^{3}}\mathrm{d}z\varleq C\frac{s}{\eta% _{\ell}^{k+3}}\varleq\frac{C}{\eta_{\ell}^{k+1}}\frac{s}{s^{2}+(x-y)^{2}} (5.14)

because \eta_{\ell}\vargeq s and \eta_{\ell}\vargeq c|x-y|. ∎

We now derive the following Hölder estimate for p_{\eta_{\ell}}.

###### Lemma 5.2.

For any 0\varleq\alpha<1 and 0\varleq t\varleq\eta_{\ell} we have

 \left|p_{\eta_{\ell}}(t,x,y)-p_{\eta_{\ell}}(t,z,y)\right|\varleq\frac{C|x-z|^% {\alpha}}{t^{1+\alpha}}. (5.15)

Proof.  We have

 p_{\eta_{\ell}}(t,x,y)=p_{\Lambda}(t,x,y)+\int_{0}^{t}(\mathrm{e}^{(t-s)% \Lambda}g(s))(x)\mathrm{d}s (5.16)

where we denoted g(s,w)=\left[(K-\Lambda){f_{\eta_{\ell}}}(s)\right](w). The first term satifies the desired bound so we nned only estimate the second term. From (5.6) and (5.10) we have the estimate

 |g(s,w)|\varleq\frac{C}{\eta_{\ell}}\left(\frac{s}{s^{2}+w^{2}}+\frac{\eta_{% \ell}}{\eta_{\ell}^{2}+w^{2}}\right). (5.17)

We estimate

 \displaystyle\left|(\mathrm{e}^{(t-s)\Lambda}g(s))(x)-(\mathrm{e}^{(t-s)% \Lambda}g(s))(z)\right| \displaystyle\varleq \displaystyle\frac{1}{\pi}\int\left|\frac{t-s}{(t-s)^{2}+(x-w)^{2}}-\frac{t-s}% {(t-s)^{2}+(z-w)^{2}}\right|g(s,w)\mathrm{d}w \displaystyle\varleq \displaystyle C(t-s)\int\frac{|x-z|^{\alpha}(|x-w|^{2-\alpha}+|z-w|^{2-\alpha}% )}{((t-s)^{2}+(x-w)^{2})((t-s)^{2}+(z-w)^{2})}|g(s,w)|\mathrm{d}w \displaystyle\varleq \displaystyle\frac{C|x-z|^{\alpha}}{(t-s)^{\alpha}}\int\frac{t-s}{(t-s)^{2}+(x% -w)^{2}}|g(s,w)|\mathrm{d}w \displaystyle+ \displaystyle\frac{C|x-z|^{\alpha}}{(t-s)^{\alpha}}\int\frac{t-s}{(t-s)^{2}+(z% -w)^{2}}|g(s,w)|\mathrm{d}w (5.18)

Using (5.17) and the bound

 \displaystyle\int\frac{a}{a^{2}+w^{2}}\frac{b}{b^{2}+(w-c)^{2}}\mathrm{d}w% \varleq C\frac{b\vee a}{(b\vee a)^{2}+c^{2}} (5.19)

we get

 \left|(\mathrm{e}^{(t-s)\Lambda}g(s))(x)-(\mathrm{e}^{(t-s)\Lambda}g(s))(z)% \right|\varleq\frac{C|x-z|^{\alpha}}{(t-s)^{\alpha}}\frac{1}{t}. (5.20)

The claim follows after integrating in s. ∎

We now prove the following.

###### Lemma 5.3.

Let 0<\alpha<1. We have,

 \left|\partial_{x}p_{\eta_{\ell}}(t,x,y)\right|\varleq C\left(\frac{1+|\log% \delta|t/\eta_{\ell}}{t^{2}+(x-y)^{2}}+\frac{\delta^{\alpha}}{t}\right) (5.21)

for any t\varleq\eta_{\ell} and 0<\delta<1.

Proof.  We have

 \partial_{x}p_{\eta_{\ell}}(t,x,y)=\partial_{x}p_{\Lambda}(t,x,y)-\int_{0}^{t}% \mathrm{e}^{(t-s)\Lambda}\partial_{x}[(K-\lambda){f_{\eta_{\ell}}}](x)\mathrm{% d}s. (5.22)

We write

 \displaystyle\mathrm{e}^{(t-s)\Lambda}\partial_{x}[(K-\lambda){f_{\eta_{\ell}}% }](x) \displaystyle=-\int\left(\partial_{w}\frac{t-s}{(t-s)^{2}+(w-x)^{2}}\right){f_% {\eta_{\ell}}}(w)\mathrm{d}w (5.23) \displaystyle+\int\frac{t-s}{(t-s)^{2}+(w-x)^{2}}g_{1}(s,w)\mathrm{d}w (5.24)

where

 g_{1}(s,w):=\int_{|w-z|\vargeq\eta_{\ell}}\frac{1}{(w-z)^{3}}{f_{\eta_{\ell}}}% (z)\mathrm{d}z. (5.25)

From (5.10) we have

 |g_{1}(s,w)|\varleq\frac{C}{\eta_{\ell}^{2}}\left(\frac{s}{s^{2}+(w-y)^{2}}+% \frac{\eta_{\ell}}{\eta_{\ell}^{2}+(w-y)^{2}}\right). (5.26)

For (5.24) we have

 \displaystyle\left|\int\frac{t-s}{(t-s)^{2}+(w-x)^{2}}g_{1}(s,w)\mathrm{d}w% \right|\varleq\frac{C}{\eta_{\ell}^{2}}\int\frac{t-s}{(t-s)^{2}+(w-(x-y))^{2}}% \left(\frac{s}{s^{2}+w^{2}}+\frac{\eta_{\ell}}{\eta_{\ell}^{2}+w^{2}}\right)% \mathrm{d}w. (5.27)

Using (5.19) and t\varleq\eta_{\ell} one easily concludes

 \frac{C}{\eta_{\ell}^{2}}\int\frac{t-s}{(t-s)^{2}+(w-(x-y))^{2}}\left(\frac{s}% {s^{2}+w^{2}}+\frac{\eta_{\ell}}{\eta_{\ell}^{2}+w^{2}}\right)\mathrm{d}w% \varleq\frac{C}{t}\frac{1}{t^{2}+(x-y)^{2}}. (5.28)

For the term (5.23) we first note that

 \displaystyle\int\partial_{w}\frac{t-s}{(t-s)^{2}+(w-x)^{2}}f_{\eta_{\ell}}(s,% w)\,\mathrm{d}w=\int\partial_{w}\frac{t-s}{(t-s)^{2}+(w-x)^{2}}(f_{\eta_{\ell}% }(s,w)-f_{\eta_{\ell}}(s,x))\,\mathrm{d}w.

Hence, using (5.6) and Lemma 5.2 we can estimate

 \displaystyle\left|\int_{0}^{t}\int\partial_{w}\frac{t-s}{(t-s)^{2}+(w-x)^{2}}% f_{\eta_{\ell}}(s,w)\,\mathrm{d}w\mathrm{d}s\right| \displaystyle\varleq C\int_{0}^{t(1-\delta)}\int\frac{(t-s)|x-w|}{((t-s)^{2}+(% w-x)^{2})^{2}}\frac{s}{s^{2}+(w-y)^{2}}\,\mathrm{d}w\mathrm{d}s (5.29) \displaystyle\ +C\int_{t(1-\delta)}^{t}\int\frac{(t-s)|x-w|^{1+\alpha}}{((t-s)% ^{2}+(w-x)^{2})^{2}}s^{-1-\alpha}\,\mathrm{d}w\mathrm{d}s (5.30)

For (5.29), we use (5.19) to obtain

 \displaystyle\int_{0}^{t(1-\delta)}\int\frac{(t-s)|x-w|}{((t-s)^{2}+(w-x)^{2})% ^{2}}\frac{s}{s^{2}+(w-y)^{2}}\,\mathrm{d}w\mathrm{d}s \displaystyle\varleq\int_{0}^{t(1-\delta)}\frac{1}{t-s}\frac{t-s}{(t-s)^{2}+(w% -x)^{2}}\frac{s}{s^{2}+(w-y)^{2}}\,\mathrm{d}w\mathrm{d}s \displaystyle\varleq C\frac{t}{t^{2}+(x-y)^{2}}\int_{0}^{t(1-\delta)}\frac{1}{% t-s} \displaystyle\varleq C|\log{\delta}|\cdot\frac{t}{t^{2}+(x-y)^{2}}.

For (5.30), we use

 \frac{|w-x|^{1+\alpha}}{(t-s)^{2}+(w-x)^{2}}\leq\frac{1}{(t-s)^{1-\alpha}}

to obtain

 \displaystyle\int_{t(1-\delta)}^{t}\int\frac{(t-s)|x-w|^{1+\alpha}}{((t-s)^{2}% +(w-x)^{2})^{2}}s^{-1-\alpha}\,\mathrm{d}w\mathrm{d}s \displaystyle\leq\int_{t(1-\delta)}^{t}(t-s)^{-1+\alpha}\int\frac{t-s}{(t-s)^{% 2}+(w-x)^{2}}s^{-1-\alpha}\,\mathrm{d}w\mathrm{d}s \displaystyle\leq\frac{C}{1-\alpha}\frac{\delta^{\alpha}}{t}.

This yields the claim. ∎

We now can conclude with estimates of the spatial derivatives of p_{\eta_{\ell}}(t,x,y).

###### Theorem 5.4.

Fix D_{1}>0 and D_{2}>0 and \delta>0. Let \eta_{\ell}=N^{\omega_{\ell}}/N for each k we have

 \left|\partial_{x}^{k}p_{\eta_{\ell}}(t,x,y)\right|\varleq\frac{C}{t^{k}}\frac% {t}{t^{2}+(x-y)^{2}}+N^{-D_{2}}, (5.31)

for N^{-D_{1}}\varleq t\varleq N^{-\delta}\eta_{\ell}. For |x-y|>N^{\varepsilon}\eta_{\ell} we have

 \left|\partial_{x}^{k}p_{\eta_{\ell}}(t,x,y)\right|\varleq N^{-D_{2}}. (5.32)

Proof.  The bound (5.31) for k=1 follows by taking \delta=N^{-D} in (5.21). For k\vargeq 2 we have by the Chapman-Kolmogorov equation and translation invariance:

 \partial_{x}^{(k-1)}p_{\eta_{\ell}}(t,x,y)=\int p(t/2,x,w)p^{(k-1)}(t/2,w,y)\,% \mathrm{d}y.

Differentiating once more, we have

 \partial_{x}^{(k)}p_{\eta_{\ell}}(t,x,y)=\int\partial_{x}p(t/2,x,w)p^{(k-1)}(t% /2,w,y)\,\mathrm{d}y. (5.33)

The bounds (5.31) for higher k then follow by strong induction and (5.19).

For (5.32) we use translation invariance and the Chapman-Kolmogorov equation to write for x>0,

 \displaystyle\partial_{x}p_{\eta_{\ell}}(t,x,0) \displaystyle=\int\partial_{w}p_{\eta_{\ell}}(t/2,x,w)p_{\eta_{\ell}}(t/2,w,0)% \mathrm{d}w \displaystyle=\int_{w\varleq x/2}\partial_{w}p_{\eta_{\ell}}(t/2,x,w)p_{\eta_{% \ell}}(t/2,w,0)\mathrm{d}w+\int_{w>x/2}\partial_{w}p_{\eta_{\ell}}(t/2,x,w)p_{% \eta_{\ell}}(t/2,w,0)\mathrm{d}w \displaystyle=-\int_{w\varleq x/2}p_{\eta_{\ell}}(t/2,x,w)\partial_{w}p_{\eta_% {\ell}}(t/2,w,0)\mathrm{d}w+\int_{w>x/2}\partial_{w}p_{\eta_{\ell}}(t/2,x,w)p_% {\eta_{\ell}}(t/2,w,0)\mathrm{d}w \displaystyle+p_{\eta_{\ell}}(t/2,x,x/2)p_{\eta_{\ell}}(t/2,x/2,0). (5.34)

If |x|>N^{\varepsilon}\eta_{\ell} we see that each term is O(N^{-D}) using (5.32). The higher order derivatives can be handled similarly. ∎

Finally we handle the time derivative.

###### Theorem 5.5.

Fix D_{1},D_{2}>0 and \delta>0. For N^{-D_{1}}\varleq t\varleq N^{-\delta}\eta_{\ell} we have

 \left|\partial_{t}p_{\eta_{\ell}}(t,x,y)\right|\varleq\frac{C}{t^{2}+(x-y)^{2}% }+N^{-D_{2}}. (5.35)

Proof.  We write

 \displaystyle\partial_{t}p_{\eta_{\ell}}(t,x,0)=\int_{|x-z|\varleq\eta_{\ell}}% \frac{p_{\eta_{\ell}}(t,x,0)-p_{\eta_{\ell}}(t,z,0)}{(x-z)^{2}}\mathrm{d}z \displaystyle=\int_{|x-z|\varleq t}\frac{p_{\eta_{\ell}}(t,x,0)-p_{\eta_{\ell}% }(t,z,0)}{(x-z)^{2}}\mathrm{d}z (5.36) \displaystyle+\int_{\eta_{\ell}>|x-z|>t}\frac{p_{\eta_{\ell}}(t,x,0)-p_{\eta_{% \ell}}(t,z,0)}{(x-z)^{2}}\mathrm{d}z (5.37)

An argument similar to the proof of Lemma 5.1 yields

 \displaystyle\left|\int_{\eta_{\ell}>|x-z|>t}\frac{p_{\eta_{\ell}}(t,x,0)-p_{% \eta_{\ell}}(t,z,0)}{(x-z)^{2}}\mathrm{d}z\right|\varleq C\int_{|x-z|>t}\frac{% 1}{(x-z)^{2}}\left(\frac{t}{t^{2}+x^{2}}+\frac{t}{t^{2}+z^{2}}\right)\mathrm{d% }z\varleq\frac{C}{t^{2}+x^{2}}. (5.38)

We now turn to (5.36). Note that

 \sup_{|u-x|\varleq t}\frac{1}{t^{2}+u^{2}}\varleq\frac{C}{t^{2}+x^{2}}. (5.39)

Hence by a second order Taylor expansion,

 \displaystyle\int_{|x-z|\varleq t}\frac{p_{\eta_{\ell}}(t,x,0)-p_{\eta_{\ell}}% (t,z,0)}{(x-z)^{2}}\mathrm{d}z \displaystyle= \displaystyle\int_{|x-z|\varleq t}\frac{\partial_{x}p_{\eta_{\ell}}(t,x,0)}{(x% -z)}+\mathcal{O}\left(\frac{1}{t}\frac{1}{t^{2}+x^{2}}+N^{-D}\right)\mathrm{d}% z=\mathcal{O}\left(\frac{1}{t^{2}+x^{2}}+N^{-D}\right). (5.40)

This yields the claim. ∎

## 6 Mesoscopic linear statistics

Let \varphi_{N} be a sequence of twice differentiable functions such that:

 \displaystyle\|\varphi_{N}\|_{L^{\infty}},\|\varphi_{N}^{\prime}\|_{L^{1}}\leq C, (6.1) \displaystyle\mathrm{supp}\varphi_{N}^{\prime}(x)\cap[-t^{1/4},t^{1/4}]\subset% (-t_{1}N^{r},t_{1}N^{r}), (6.2)

for some 0<r<\omega_{1}/100.

We will assume throughout that t_{1}=N^{\omega_{1}}/N. As for the parameter t, we assume t_{1}N^{r}\ll t\ll 1, and we will denote t=N^{\omega_{0}}/N.

Finally, since the spectrum of H_{t} is contained in -N^{C_{V}}-1,N^{C_{V}}+1 with overwhelming probability, there is no loss of generality in making the following assumption on the support of \varphi_{N}:

 \mathrm{supp}\varphi_{N}\subset[-N^{C_{V}}-2,N^{C_{V}}+2]. (6.3)

We can now state the main result of this section. After introducing some notation in Section 6.1 we give an outline of the proof in Section 6.2. The majority of the remainder of Section 6 is concerned with the proof.

###### Theorem 6.1.

Let \varphi_{N} be a sequence of real-valued C^{2}(\mathbb{R}) functions satisfying (6.1), (6.2), (6.3) in addition to the following growth conditions on the derivatives:

and

 \int\int\left(\frac{\varphi_{N}(x)-\varphi_{N}(y)}{x-y}\right)^{2}\,\mathrm{d}% x\mathrm{d}y\geq c. (6.5)

Let the parameters t=N^{\omega_{0}}/N and t_{1}=N^{\omega_{1}}/N satisfy

 \displaystyle\omega_{0}>\omega_{1}>\omega_{0}/2 (6.6)

Then, uniformly in |x|\leq N^{\omega_{1}/8-\omega_{0}/16},

 \mathbb{E}[e^{ix[(\mathrm{tr}\varphi_{N})(H_{t})-\mathbb{E}\mathrm{tr}\varphi_% {N}(H_{t})]}]=\exp\left(-\frac{x^{2}}{2}V(\varphi_{N})\right)+\mathcal{O}_{% \prec}(N^{\omega_{0}/4-\omega_{1}/2}). (6.7)

Here V(\varphi_{N}) is a quadratic functional in \varphi_{N} such that

 \begin{split}\displaystyle V(\varphi_{N})&\displaystyle=-\frac{1}{\pi^{2}}\int% _{-Ct}^{Ct}\varphi_{N}(\tau)(H\varphi_{N}^{\prime})(\tau)\,\mathrm{d}s\mathrm{% d}\tau+\mathcal{O}(1)\\ &\displaystyle=\frac{1}{2\pi^{2}}\int_{-Ct}^{Ct}\int_{-Ct}^{Ct}\left(\frac{% \varphi_{N}(\tau)-\varphi_{N}(s)}{\tau-s}\right)^{2}\,\mathrm{d}s\mathrm{d}% \tau+\mathcal{O}(1).\end{split} (6.8)

Here C>0 is some (small) constant, and H denotes the Hilbert transform (see 6.130). In particular,

 V(\varphi_{N})\geq c\log t/t_{1}+\mathcal{O}(1)

if \varphi_{N}=\int_{0}^{x}\chi(y/(t_{1}N^{\alpha}))p_{t_{1}}(0,y)\,\mathrm{d}y.

If \mathrm{supp}\varphi_{N}\subset(-N^{r}t_{1},N^{r}t_{1}), then we have the more precise evaluation:

 \begin{split}\displaystyle V(\varphi_{N})&\displaystyle=-\frac{1}{\pi^{2}}\int% \varphi_{N}(\tau)(H\varphi_{N}^{\prime})(\tau)\,\mathrm{d}\tau+\mathcal{O}(N^{% \omega_{0}/2-\omega_{1}}N^{2r})\\ &\displaystyle=\frac{1}{2\pi^{2}}\int\int\left(\frac{\varphi_{N}(\tau)-\varphi% _{N}(s)}{\tau-s}\right)^{2}\,\mathrm{d}s\mathrm{d}\tau+\mathcal{O}(N^{\omega_{% 0}/2-\omega_{1}}N^{2r}).\end{split} (6.9)

Remark. We make several comments concerning Theorem 6.1 and the many conditions in the statement.

1. The first inequality of (6.6) ensures that the scale of the function is smaller than the time scale of DBM. See the remark after the statement of Theorem 2.5.

2. The second inequality of (6.6) is technical, but removing it requires substantial modification of some of the estimates below. It is used in particular to simplify the handling of some error terms in the proof of Proposition 6.5, which is key in deriving the theorem.

3. The condition (6.5) ensures the limiting random variable is non-degenerate, that is, its variance is bounded below. It is used only in the final integration at (6.55). Our method can be extended to cover the case of vanishing variance, but we will have no need for such an extension.

4. A typical situation in which Theorem 6.1 holds with the approximation (6.9) is when

 \varphi_{N}(x)=\varphi(x/t_{1}),

where \varphi is some smooth function either compactly supported or vanishing as |x|\to\infty. This setting has been studied extensively in the random matrix theory literature (see for example [31], [59], [44]), and is typically what is being referred to when one speaks of “linear statistics of mesoscopic observables”. The more general theorem above is essential for the main result of this paper.

5. If the functions \varphi_{N} do not have spatial decay, the variance of the linear statistics grows logarithmically. This should be compared to the well-known fact that the variance of the number of eigenvalues in an interval grows like \log N. See [29, 62]. A function \varphi_{N} with “large” (compared to t) support, but whose derivative is supported in a region of size t_{1} is, up to a linear transformation, an approximation on scale t_{1} of an indicator function.

Concerning the last remark above, we note that the fact that we allow for non-compactly supported functions (which is required for the proof) causes substantial technical difficulties. Many alternative approaches would be viable if it sufficed to consider compactly supported \varphi.

### 6.1 Notation for resolvents

A central role will be played by the resolvent matrix

 G(z)=(H_{t}-z)^{-1}, (6.10)

where z=\tau+\mathrm{i}\eta\in\mathbb{C}. The normalized trace of G is denoted by m_{N}(z):

 m_{N}(z)=\frac{1}{N}\mathrm{tr}G(z). (6.11)

The latter quantity closely approximates the Stieltjes transform m_{\mathrm{fc},t}(z) of the deformed semicircle law.

Let H^{(j)} be the (j,j)-submatrix of H_{t}, that is, the (N-1)\times(N-1) matrix obtained from the Wigner matrix H by removing the jth row and column. We introduce the following notation for the resolvent of H^{(j)} and its normalized trace:

Following [64], we reserve special symbols for two quantities involving G and G^{(j)} which will play a role in the computations to come. First, we denote, for j=1,\ldots,N,

 A_{j}=A_{j}(z):=-\frac{1}{G_{jj}(z)}. (6.13)

Next, we let h^{(j)}:=(h_{ji})_{1\leq i\leq N}, and then define

 B_{j}=B_{j}(z):=\langle(G^{(j)}(z))^{2}h^{(j)},h^{(j)}\rangle, (6.14)

where \langle\mathbf{u},\mathbf{v}\rangle denotes the inner product of the vectors \mathbf{u},\mathbf{v}\in\mathbb{C}^{N}. The importance of these quantities for us comes mainly through the identity (6.34).

Following [55], we also define

 g_{i}(z)=\frac{1}{V_{i}-z-tm_{\mathrm{fc},t}(z)}, (6.15)

so that

 m_{\mathrm{fc},t}(z)=\frac{1}{N}\sum_{i=1}^{N}g_{i}(z).

Finally, we define

We will often deal with centered random variables. For a random variable X with \mathbb{E}|X|<\infty, we denote by

 X^{\circ}:=X-\mathbb{E}[X], (6.16)

the corresponding centered random variable.

### 6.2 Outline of the proof

Let us now outline the strategy of proof and provide a guide for the reader. The main idea is to compute the derivative of the characteristic function of the random variable

 Z:=\mathrm{tr}\varphi_{N}(H)-\mathbb{E}\mathrm{tr}\varphi_{N}(H).

This approach has been previously applied to linear statistics of random matrices by Shcherbina [64]. Denoting \psi(x)=\mathbb{E}[\mathrm{e}^{ixZ}], we have

 \psi^{\prime}(x)=-\mathrm{i}\mathbb{E}[Z\mathrm{e}^{\mathrm{i}xZ}], (6.17)

If we can show that the left side is close to -xV(\varphi_{N})\times\psi(x), then it follows by direct integration in x that Z is approximately normal.

Next, the problem of computing \mathbb{E}[Ze^{\mathrm{i}xZ}] is reduced to computations involving the matrix H (more precisely, the resolvent G(z)) through the Helffer-Sjöstrand representation (6.40):

 \mathbb{E}[Ze^{\mathrm{i}xZ}]=\int_{\mathbb{C}}\mathbb{E}[\mathrm{tr}G(\tau+% \mathrm{i}\eta)e(x)^{\circ}]\partial_{z}\tilde{\varphi}(z)\,\mathrm{d}z.

Here

 e^{\circ}(x)=\exp(\mathrm{i}xZ)-\mathbb{E}[\exp(\mathrm{i}xZ)], (6.18)

and the function \tilde{\varphi_{N}}(z) is a quasi-analytic extension of \varphi_{N} to the complex plane \mathbb{C},

 \tilde{\varphi}_{N}(x+\mathrm{i}\eta)=\chi(\eta)(\varphi_{N}(x)+\mathrm{i}\eta% \varphi_{N}^{\prime}(x)) (6.19)

where \chi(\eta) is a smooth compactly supported cut-off function equal to 1 in neighborhood of 0. The quantity \mathbb{E}[Ze^{\mathrm{i}xZ}] is then decomposed into two pieces:

 \displaystyle\int_{\mathbb{C}}\mathbb{E}[\mathrm{tr}G(\tau+\mathrm{i}\eta)e^{% \circ}]\partial_{z}\tilde{\varphi}_{N}(z)\,\mathrm{d}z \displaystyle=\int\partial_{z}\tilde{\varphi}_{N}(z)(T_{1}(z)+T_{2}(z))\,% \mathrm{d}z, \displaystyle T_{1}(\tau,\eta) \displaystyle:=\mathbb{E}\sum_{j=1}^{N}\left[G_{jj}^{\circ}(\tau+\mathrm{i}% \eta)e_{j}^{\circ}\right], \displaystyle T_{2}(\tau,\eta) \displaystyle:=\mathbb{E}\sum_{j=1}^{N}\left[G_{jj}^{\circ}(\tau+\mathrm{i}% \eta)(e-e_{j})\right].

Above, e_{j} is the same as e, but with the minor H^{(j)} replacing H. Through careful resolvent expansions, it will be found that the integral involving T_{1} is close to a multiple of \mathbb{E}[Ze^{\mathrm{i}xZ}] itself:

 \int\partial_{z}\tilde{\varphi}_{N}(z)T_{1}(z)\,\mathrm{d}z=t\int\partial_{z}% \tilde{\varphi}_{N}(z)\tilde{R}_{2}(z)\cdot\mathbb{E}[\mathrm{tr}G(\tau+% \mathrm{i}\eta)e(x)^{\circ}]\,\mathrm{d}z+\text{error}.

This relatively straightforward computation appears in Section 6.5. The main input here is that the dominant contribution to the fluctuations of G_{jj} are caused by the jth row and column of H. Since e_{j} is independent of these matrix entries, a resolvent expansion based on the Schur complement formula allows for the calculation of the expectation over the jth row and column (i.e, the expectation conditional on H^{(j)}), ultimately leading to the above expression.

The computation of T_{2} is more involved. It results in the appearance of a deterministic kernel depending on m_{\mathrm{fc},t} which will ultimately generate the covariance kernel \sqrt{-\Delta} (the square root of the Laplacian having integral kernel (\cdot-y)^{-2}) appearing in the statement of the theorem. Part of this is the statement of Proposition 6.5, which is:

 \displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{T_% {2}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z (6.20) \displaystyle= \displaystyle-\frac{2\mathrm{i}x}{\pi}\mathbb{E}[e(x)]\int_{\Omega_{N}}\int_{% \Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})\frac{1}{1-tR_{2}(z)}S_{2,1}(z,z^{\prime})\,\mathrm{d}% z\mathrm{d}z^{\prime} (6.21) \displaystyle+\frac{2\mathrm{i}x}{\pi}\mathbb{E}[e(x)]\int_{\Omega_{N}}\int_{% \Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})\frac{1}{1-tR_{2}(z)}(S_{2,2}(z,z^{\prime})+S_{2,3}(z,% z^{\prime}))\,\mathrm{d}z\mathrm{d}z^{\prime}+\text{error}. (6.22)

The kernels S_{2,1}, S_{2,2}, S_{2,3} are defined in (6.51), (6.52), (6.53). They are deterministic functions of the initial data V and m_{\mathrm{fc},t}. We remark that the transition between Gaussian statistics with a universal variance profile when t\gg t_{1} and a distribution depending on V when t\ll t_{1} alluded to in the introduction to this paper can essentially be understood by looking at the behavior of the quantities S_{1,2}, S_{2,2} appearing in (6.51) and (6.52) when t depends on N. The computation of T_{2} appears in Sections 6.6-6.8.

We now summarize the proof of Proposition 6.5. In the initial step, we Taylor expand the difference e-e_{j} in powers of \mathrm{i}x(\mathrm{tr}\varphi_{N}(H)-\mathrm{tr}\varphi_{N}(H^{(j)}))^{\circ}. Already, the quadratic term will be negligible. We will use the Helffer-Sjöstrand formula to evaluate the linear term. This is the source of the second integration over \mathbb{C} and the term \partial_{z^{\prime}}\varphi_{N}(z^{\prime}) in (6.20) above, as well as the prefactor x which must appear on the RHS of (6.17) in order to conclude the Gaussian statistics. A resolvent expansion yields the following expression for T_{2}:

 \displaystyle T_{2}= \displaystyle-\sum_{j=1}^{N}\frac{1}{\mathbb{E}[A_{j}(\tau+\mathrm{i}\eta)]^{2% }}\frac{\mathrm{i}x}{\pi}\int\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})% \mathbb{E}\left[e_{j}(x)\left(N[m_{N}^{(j)}-m_{N}](s+\mathrm{i}\eta^{\prime})% \right)^{\circ}A_{j}(\tau+\mathrm{i}\eta)^{\circ}\right]\mathrm{d}z^{\prime} (6.23) \displaystyle+\text{error}.

where A_{j}(z)=-(G_{jj}(z))^{-1}. In Section 6.6 we expand the main term in (6.23). The leading terms resulting in S_{2,1}, S_{2,2}, S_{2,3} are computed in Sections 6.7 and 6.6, while the error terms are estimated in Section 6.8.

In Section 6.9, we consider the quadratic expression in (6.20), ultimately deriving the simplified expression (6.8) for the asymptotic variance. This expression is approximately equal to a constant factor times the Sobolev norm (\varphi,\sqrt{-\Delta}\varphi)^{2}. The reader will note that this section could be drastically simplified if we were dealing with functions with compact support. After some simple manipulations involving m_{\mathrm{fc},t} (see Proposition 6.12), the main work is in transforming, up to some errors, the area integrals over \mathbb{C} which appear in (6.127), (6.128) into line integrals over \mathbb{R} (using Green’s theorem) and isolating the main terms. It is found that ultimately the only non-vanishing contribution to the variance comes from the expression S_{2,1} (6.127). Once the error terms are dealt with the variance kernel (6.8) comes out of some essentially exact computations involving m_{\mathrm{fc},t}.

#### 6.2.1 A simple example

In order to illustrate the guiding principles of these calculations, let us consider the simpler case of the Stieltjes transform instead of the general test function \varphi above (in fact, in a moment we will just consider the calculation of its expectation). This central limit theorem is best understood as an extension of the local law. Recall that the local law states that |m_{N}(z)-m_{\mathrm{fc},t}(z)|\varleq N^{\varepsilon}/(N\eta) with overwhelming probability. The (imaginary and real parts of the) quantity (N\eta)(m_{N}(z)-m_{\mathrm{fc},t}(z)) is expected to satisfy a central limit theorem. In order to prove the local law, one uses the Schur complement formula and after some simplification arrives at,

 m_{N}(z)=\frac{1}{N}\sum_{j=1}^{N}G_{jj}(z)=\frac{1}{N}\sum_{j=1}^{N}\frac{1}{% V_{j}-z-tm_{\mathrm{fc},t}(z)+\varepsilon_{j}} (6.24)

where \varepsilon_{j} is an error term. The local law can be viewed as finding large deviations estimates for the error terms \varepsilon_{j}. The central limit theorem can be viewed as a more careful consideration of the error terms \varepsilon_{j}. This analysis is aided by the fact that one can, e.g., consider moments of the quantity (N\eta)(m_{N}(z)-m_{\mathrm{fc},t}(z)) which results in one only having to calculate the first few moments of \varepsilon_{j}.

As a simple example, let us consider the task of calculating the expectation of (N\eta)(m_{N}(z)-m_{\mathrm{fc},t}(z)) up to o(1) errors. One can Taylor expand each term on the RHS of (6.24) in powers of \varepsilon_{j}. The large deviations estimates on \varepsilon_{j} are sufficient to truncate this expansion after a few terms (in this case the 3rd order). The form of \varepsilon_{j} is roughly \varepsilon_{j}=\sum_{k,l\neq j}h_{jk}(G^{(j)}_{kl}-N^{-1}\delta_{kl})h_{lj}. The difference between this and the local law is that instead of proving large deviations estimates on \varepsilon_{j}, we can use them as a starting point in order to calculate a few moments of \varepsilon_{j}, thanks to the presence of the expectation infront of \mathbb{E}[(N\eta)(m_{N}(z)-m_{\mathrm{fc},t}(z))]. The moments of \varepsilon_{j} are most easily calculated by first taking the partial expectation over the jth row of H. The term \mathbb{E}[\varepsilon_{j}^{2}] is seen to be negligible and \mathbb{E}[\varepsilon_{j}] gives \mathbb{E}[m_{N}^{(j)}-m_{\mathrm{fc},t}]. This expression can be rewritten as \mathbb{E}[m_{N}^{(j)}-m_{\mathrm{fc},t}]=\mathbb{E}[m_{N}-m_{\mathrm{fc},t}]+% \mathbb{E}[m_{N}^{(j)}-m_{N}]. The first expression is the analogue of T_{1} described above, and is the expression we wished to calculate in the first place. It appears with a coefficient (in this case R_{2}) and is moved back over to the LHS of the equation. The second term is the analogue of T_{2}, and an algebraic identity (this is (6.34) below) together with a further resolvent expansion allows for its calculation.

In a full proof of a central limit theorem, one will be calculating the expectation of (N\eta)(m_{N}-m_{\mathrm{fc},t}) times either its characteristic function as above, or a monomial in it and its conjugate if one is proceeding by the method of moments. In these cases, this factor must also be expanded around the corresponding expression involving H^{(j)}, which coordinates well with the expansion of G_{jj} using the Schur complement formula.

In summary, the local law determines m_{N} down to N^{\varepsilon}/(N\eta). In order to remove the N^{\varepsilon} factor, we proceed similarly the proof of the local law based around resolvent expansions, except that we use the independence structure of the matrix ensemble to calculate expectations of the first order of error terms.

### 6.3 Estimates for A_{j} and B_{j}

The following definition will be useful.

###### Definition 6.2 (Stochastic Domination).

Let

be two families of nonnegative random variables, where U^{(N)} is a possibly N-dependent parameter set. We say that X is stochastically dominated by Y, uniformly in u, if for all small \epsilon>0 and all (large) D we have

 \sup_{u\in U^{(N)}}\mathbb{P}(X^{(N)}>N^{\epsilon}Y^{(N)}(u))\leq N^{-D}

for all N\geq N_{0}(\epsilon,D). If X is stochastically dominated by Y, uniformly in u, we write

 X\prec Y. (6.25)

For complex valued Y, we write Y=\mathcal{O}_{\prec}(X) if |Y|\prec X.

Recall the definition of \mathcal{D}_{\epsilon,q}. We define \Omega_{N} as the union of this region with its reflection about the real axis, with a choice \epsilon=\xi to be determined.

 \begin{split}\displaystyle\Omega_{N}&\displaystyle=(\mathcal{D}_{\xi,q}\cup% \overline{\mathcal{D}_{\xi,q}})\\ &\displaystyle=\left\{z=E+\mathrm{i}\eta:E\in\mathcal{I}_{q},N^{10C_{V}}% \vargeq|\eta|\vargeq N^{\xi}/N\right\}\cup\left\{z:E+\mathrm{i}\eta:|E|\varleq N% ^{2C_{V}},N^{C_{V}}\vargeq|\eta|\vargeq c\right\}.\end{split} (6.26)

Since m_{N}(\bar{z})=\overline{m_{N}(z)} and m_{\mathrm{fc},t}(\bar{z})=\overline{m_{\mathrm{fc},t}(z)}, the local law extends to z\in\Omega_{N}.

The following estimates for the quantities A and B defined in (6.13) and \eqref{eqn: Bdef}:

###### Theorem 6.3.

We have, uniformly in z\in\Omega_{N},

 \displaystyle\mathbb{E}_{j}A_{j}(z) \displaystyle=z+tm_{\mathrm{fc},t}(z)-V_{j}+\mathcal{O}_{\prec}(t(N|\eta|)^{-1% }), (6.27) \displaystyle A_{j}^{\circ} \displaystyle=\mathcal{O}_{\prec}(\sqrt{t}N^{-1/2}+t(N|\eta|)^{-1/2})) (6.28) \displaystyle B_{j}(z) \displaystyle=t\partial_{z}m_{\mathrm{fc},t}(z)+\mathcal{O}_{\prec}(t|\eta|^{-% 1}(N|\eta|)^{-1/2}). (6.29)

Proof.   Recall the definition of A_{j}(x+\mathrm{i}\eta) above. By the Schur complement formula [55, Lemma 7.7],

 \begin{split}\displaystyle A_{j}(z)=z-h_{jj}+\langle G^{(j)}h^{(j)},h^{(j)}% \rangle.\end{split} (6.30)

Taking the partial expectation over h^{(j)}=(h_{ji})_{i\neq j}, we have

 \mathbb{E}_{j}A_{j}(z)=z-V_{j}+\frac{t}{N}\mathrm{tr}G^{(j)}(z). (6.31)

Using the local law, we obtain

 \mathbb{E}_{j}A_{j}(z)=z-V_{j}+tm_{\mathrm{fc},t}(z)+\mathcal{O}_{\prec}(t(N|% \eta|)^{-1}),

which is (6.27).

The estimate (6.28) is proved in [55, Lemma 7.9] using the local law.

For (6.29), note that

 B_{j}=\langle(G^{(j)})^{2}h^{(j)},h^{(j)}\rangle=\partial_{z}\langle G^{(j)}h^% {(j)},h^{(j)}\rangle.

Taking the expectation with respect to h^{(j)} first, and then using the local law, we find

 \begin{split}\displaystyle\mathbb{E}[B_{j}]&\displaystyle=t\mathbb{E}[\partial% _{z}m_{N}^{(j)}]\\ &\displaystyle=t\partial_{z}\oint_{|z-\zeta|=\eta/2}\frac{\mathbb{E}[m_{N}^{(j% )}](\zeta)}{z-\zeta}\,\mathrm{d}\zeta\\ &\displaystyle=t\partial_{z}m_{\mathrm{fc},t}(z)+\mathcal{O}_{\prec}(t|\eta|^{% -1}(N|\eta|)^{-1}).\end{split} (6.32)

For the second moment, we have

 \displaystyle\mathbb{E}[|B_{j}^{\circ}|^{2}] \displaystyle=\mathbb{E}[\left|\sum^{(j)}_{i,r,k}G^{(j)}_{ir}G^{(j)}_{rk}(h_{% ji}h_{kj}-\delta_{ik}tN^{-1})\right|^{2}]+\mathcal{O}(t^{2}|\eta|^{-2}(N|\eta|% )^{-2}) \displaystyle=\mathbb{E}\sum^{(j)}_{i,l,k,r,m,n}G^{(j)}_{il}G^{(j)}_{lk}\bar{G% }^{(j)}_{rm}\bar{G}^{(j)}_{mn}\mathbb{E}_{j}[(h_{ji}h_{kj}-N^{-1}t\delta_{ik})% (h_{jr}h_{nj}-N^{-1}t\delta_{rn})]+\mathcal{O}(t(N|\eta|)^{-2}) \displaystyle=\frac{2t^{2}}{N^{2}}\mathbb{E}\mathrm{tr}|G^{(j)}|^{4}+\mathcal{% O}(t(N|\eta|)^{-2})=\mathcal{O}(t^{2}|\eta|^{-2}(N|\eta|)^{-1}). (6.33)

(6.29) now follows from the large deviation type estimates in [55, Lemma 7.7], and the local law. ∎

We will repeatedly use the identity:

 N(m_{N}-m_{N}^{(j)})=G_{jj}\left(1+\sum^{(j)}_{i,l,k}h_{ji}G^{(j)}_{il}G^{(j)}% _{lk}h_{kj}\right)=-A_{j}^{-1}(1+B_{j}). (6.34)

The following lemma collects the main estimates we need for this quantity.

###### Lemma 6.4.

Uniformly for \tau+\mathrm{i}\eta\in\Omega_{N},

 \displaystyle A_{j}^{-1}(1+B_{j}) \displaystyle=\frac{1+t\partial_{z}m_{\mathrm{fc,t}}}{\mathbb{E}[A_{j}]}+% \mathcal{O}_{\prec}(|\eta|^{-1}(N|\eta|)^{-1/2}), (6.35) \displaystyle\left((A_{j}^{-1})(1+B_{j})\right)^{\circ} \displaystyle=\frac{B_{j}^{\circ}}{\mathbb{E}[A_{j}]}-\frac{A_{j}^{\circ}(1+% \mathbb{E}B_{j})}{\mathbb{E}[A_{j}]^{2}}+\frac{1}{\mathbb{E}[A_{j}]^{2}}(A_{j}% ^{\circ}B_{j}^{\circ})^{\circ}+\frac{1}{\mathbb{E}[A_{j}]^{2}}\left(\frac{(A_{% j}^{\circ})^{2}}{A_{j}}(1+B_{j})\right)^{\circ}. (6.36)

Proof.   The first estimate follows directly from (6.28), (6.29) and the stability estimate [55, Eqn (7.8)]

 |V_{i}-z-tm_{\mathrm{fc},t}(z)|\geq c\max(t,|\eta|). (6.37)

We begin by using the expansion

 \begin{split}\displaystyle\frac{1}{A}&\displaystyle=\frac{1}{\mathbb{E}A}\cdot% \frac{1}{1+\frac{A^{\circ}}{\mathbb{E}A}}=\frac{1}{\mathbb{E}A}\left(1-\frac{A% ^{\circ}}{\mathbb{E}A}+\frac{(A^{\circ})^{2}}{(\mathbb{E}A)^{2}}-\ldots+(-1)^{% k}\frac{1}{\mathbb{E}[A]^{k}}\frac{(A^{\circ})^{k}}{1+\frac{A^{\circ}}{\mathbb% {E}A}}\right).\end{split} (6.38)

For (6.36), we expand using (6.38) with k=2,

 \begin{split}\displaystyle\left((A_{j}^{-1})(1+B_{j})\right)^{\circ}&% \displaystyle=\frac{B_{j}^{\circ}}{\mathbb{E}[A_{j}]}-\frac{1}{\mathbb{E}[A_{j% }]^{2}}\left(A_{j}^{\circ}(1+B_{j})\right)^{\circ}+\left(\frac{(A_{j}^{\circ})% ^{2}(1+B_{j})}{\mathbb{E}A_{j}+A_{j}^{\circ}}\right)^{\circ}\\ &\displaystyle=\frac{B_{j}^{\circ}}{\mathbb{E}[A_{j}]}-\frac{A_{j}^{\circ}(1+% \mathbb{E}B_{j})}{\mathbb{E}[A_{j}]^{2}}+\frac{1}{\mathbb{E}[A_{j}]^{2}}(A_{j}% ^{\circ}B_{j}^{\circ})^{\circ}+\frac{1}{\mathbb{E}[A_{j}]^{2}}\left(\frac{(A_{% j}^{\circ})^{2}}{A_{j}}(1+B_{j})\right)^{\circ}.\end{split} (6.39)

### 6.4 Computation of the characteristic function

We derive an equation for the derivative of the characteristic function of the linear statistic. Let z=\tau+\mathrm{i}\eta. Recall the definition of C_{V} in (2.3). Without loss of generality, we can assume C_{V}\geq 5. We let \chi be a smooth cut-off function such that \chi(x)=1, for |x|\leq N^{10C_{V}}-1 and \chi(x)=0, for |x|\geq N^{10C_{V}}. Next, define the almost analytic extension of \varphi_{N} to \mathbb{C}.

 \tilde{\varphi}_{N}(z)=\chi(\eta)(\varphi(\tau)+\mathrm{i}\eta\varphi_{N}^{% \prime}(\tau)).

The Helffer-Sjöstrand formula is the following representation of \varphi_{N}:

 \begin{split}\displaystyle\varphi_{N}(\lambda)&\displaystyle=\frac{1}{\pi}\int% \frac{\partial_{\bar{z}}\tilde{\varphi}_{N}(\tau+\mathrm{i}\eta)}{\lambda-\tau% -\mathrm{i}\eta}\,\mathrm{d}\tau\mathrm{d}\eta=\frac{1}{\pi}\int_{\mathbb{R}^{% 2}}\frac{\mathrm{i}\eta\varphi_{N}^{\prime\prime}(\tau)\chi(\eta)+\mathrm{i}(% \varphi_{N}(\tau)+\mathrm{i}\eta\varphi^{\prime}_{N}(\tau))\chi^{\prime}(\eta)% }{\lambda-\tau-\mathrm{i}\eta}\,\mathrm{d}\tau\mathrm{d}\eta.\end{split} (6.40)

Define

By (6.40), the derivative \psi^{\prime}(x) equals

 \displaystyle\frac{\mathrm{i}}{\pi}\int_{\mathbb{R}^{2}}(\mathrm{i}\eta\varphi% _{N}^{\prime\prime}(\tau)\chi(\eta)+\mathrm{i}(\varphi_{N}(\tau)+\mathrm{i}% \eta\varphi^{\prime}_{N}(\tau))\chi^{\prime}(\eta))E(z)\,\mathrm{d}\tau\mathrm% {d}\eta, (6.42) \displaystyle E(z):=N\mathbb{E}[e(x)(m_{N}(\tau+\mathrm{i}\eta)-\mathbb{E}m_{N% }(\tau+\mathrm{i}\eta))]. (6.43)

The rest of this section is concerned with computing E(z). Let

 e_{j}(x):=\exp\left(\mathrm{i}x\int_{\mathbb{R}^{2}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(\tau)\,\mathrm{tr}G^{(j)}(\tau+\mathrm{i}\eta)^{\circ}\,\mathrm{d% }\eta\mathrm{d}\tau\right).

We write

 \begin{split}\displaystyle-\mathrm{i}\psi^{\prime}(x)=&\displaystyle\int_{% \mathbb{C}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\mathbb{E}[\mathrm{tr}G(% \tau+\mathrm{i}\eta)e^{\circ}]\,\mathrm{d}z=\int_{\mathbb{C}}\partial_{\bar{z}% }\tilde{\varphi}_{N}\sum_{j=1}^{N}\mathbb{E}\left[G_{jj}(\tau+\mathrm{i}\eta)e% ^{\circ}\right]\,\mathrm{d}z\\ \displaystyle=&\displaystyle\int_{\mathbb{C}}\partial_{\bar{z}}\tilde{\varphi}% _{N}\ \sum_{j=1}^{N}\mathbb{E}\left[G_{jj}^{\circ}(\tau+\mathrm{i}\eta)e_{j}^{% \circ}\right]\,\mathrm{d}z+\int_{\mathbb{C}}\partial_{\bar{z}}\tilde{\varphi}_% {N}\sum_{j=1}^{N}\mathbb{E}\left[G_{jj}^{\circ}(e-e_{j})\right]\,\mathrm{d}z.% \end{split} (6.44)

In view of (6.44), we define

 \begin{split}\displaystyle T_{1}(\tau,\eta)&\displaystyle:=\mathbb{E}\sum_{j=1% }^{N}\left[G_{jj}^{\circ}(\tau+\mathrm{i}\eta)e_{j}^{\circ}\right],\qquad T_{2% }(\tau,\eta):=\mathbb{E}\sum_{j=1}^{N}\left[G_{jj}^{\circ}(\tau+\mathrm{i}\eta% )(e-e_{j})\right].\end{split} (6.45)

We compute these two terms in Propositions 6.6 and 6.5. The result of Proposition 6.6 is

 \displaystyle T_{1} \displaystyle=t\tilde{R}_{2}(z)\cdot\mathbb{E}[e(x)\mathrm{tr}G(\tau+\mathrm{i% }\eta)^{\circ}]+\mathcal{O}_{\prec}(N^{-1/2}|\eta|^{-3/2})+|x|\mathcal{O}_{% \prec}\left(|\eta|^{-1}N^{-1/2}\|\varphi_{N}^{\prime\prime}\|^{1/2}\|\varphi_{% N}^{\prime}\|_{L^{1}}^{1/2}\right).

By the definition of T_{1} and T_{2} (6.45), we have, for z\in\Omega_{N},

 \displaystyle(1-t\tilde{R}_{2}(z))\mathbb{E}[e^{\circ}(x)\mathrm{tr}G(z)] \displaystyle=T_{2}(z)+\mathcal{O}_{\prec}(N^{-1/2}|\eta|^{-3/2})+|x|\mathcal{% O}_{\prec}\left(|\eta|^{-1}N^{-1/2}\|\varphi_{N}^{\prime\prime}\|^{1/2}\|% \varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\right),

so, since |1-t\tilde{R}_{2}|\geq c by [55, Eqn (7.10)] and Proposition 6.3,

 \begin{split}\displaystyle\mathbb{E}[e^{\circ}(x)\mathrm{tr}G(z)]&% \displaystyle=\frac{T_{2}(z)}{1-t\tilde{R}_{2}(z)}+\mathcal{O}_{\prec}(N^{-1/2% }|\eta|^{-3/2})+|x|\mathcal{O}_{\prec}\left(|\eta|^{-1}N^{-1/2}\|\varphi_{N}^{% \prime\prime}\|^{1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\right).\end{split} (6.46)

Write:

 \displaystyle\int\bar{\partial}_{z}\tilde{\varphi}_{N}(z)\mathbb{E}[e^{\circ}(% x)\mathrm{tr}G(z)]\,\mathrm{d}z \displaystyle=\int_{\Omega_{N}}\bar{\partial}_{z}\tilde{\varphi}_{N}(z)\mathbb% {E}[e^{\circ}(x)\mathrm{tr}G(z)]\,\mathrm{d}z+\int_{\Omega_{N}^{c}}\bar{% \partial}_{z}\tilde{\varphi}_{N}(z)\mathbb{E}[e^{\circ}(x)\mathrm{tr}G(z)]\,% \mathrm{d}z \displaystyle=: \displaystyle I_{1}+I_{2}.

For I_{2}, note that \Omega_{N}^{c}\cap\mathrm{supp}\,\chi(\eta)\subset\{z:|\mathrm{Im}\mbox{ }z|<N% ^{-1+\xi}\}, so we have

 \begin{split}\displaystyle I_{2}&\displaystyle=2\int_{0<\eta

For I_{1}, we use (6.46):

 \displaystyle I_{1} \displaystyle=\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{T% _{2}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z (6.48) \displaystyle+\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\Delta_% {1}(z)\,\mathrm{d}z=:I_{1}^{\prime}+\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z)\Delta_{1}(z)\,\mathrm{d}z, (6.49)

where

 \Delta_{1}=\mathbb{E}[e^{\circ}(x)\mathrm{tr}G(z)]-\frac{T_{2}(z)}{1-t\tilde{R% }_{2}(z)}

is a holomorphic function in \Omega_{N} satisfying the bounds:

 \Delta_{1}=\mathcal{O}_{\prec}(N^{-1/2}|\eta|^{-3/2})+|x|\mathcal{O}_{\prec}% \left(|\eta|^{-1}N^{-1/2}\|\varphi_{N}^{\prime\prime}\|^{1/2}_{L^{1}}\|\varphi% _{N}^{\prime}\|_{L^{1}}^{1/2}\right).

Using integration by parts in \tau=\mathrm{Re}z when |\eta|\geq\|\varphi_{N}^{\prime\prime}\|_{L^{1}}^{-1} as in the proof of Lemma 6.7 (see (6.73)), it is easily shown that

 \int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\Delta_{1}\,\mathrm{d% }z=(1+|x|)\mathcal{O}_{\prec}(N^{-1/2}\log N)\|\varphi_{N}^{\prime\prime}\|^{1% /2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{3/2}. (6.50)

We compute the main term in I_{1}^{\prime}. We need an expression for T_{2}. The next proposition will be proved in the following sections.

###### Proposition 6.5.

Let

 S_{2,1}(z,z^{\prime})=\frac{t^{2}}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{\prime% })\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}% )}{z-z^{\prime}}, (6.51)
 S_{2,2}(z,z^{\prime}):=\frac{t^{2}}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{% \prime})^{2}(1+t\partial_{z}m_{\mathrm{fc},t}(z^{\prime}))\frac{m_{\mathrm{fc}% ,t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}, (6.52)

and

 S_{2,3}(z,z^{\prime}):=\frac{t}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{\prime})^% {2}(1+t\partial_{z}m_{\mathrm{fc},t}(z^{\prime})). (6.53)

The quantity I_{1}^{\prime} (6.48) is given by

Recall the definition of \psi in (6.41). By Proposition 6.5, (6.47), (6.50), we have

where

 \begin{split}\displaystyle V(\varphi_{N})&\displaystyle:=-\frac{2}{\pi^{2}}% \int_{\Omega_{N}}\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)% \partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{1-tR_{2}(z)}S_{2,1}(% z,z^{\prime})\,\mathrm{d}z\mathrm{d}z^{\prime}\\ &\displaystyle\quad+\frac{2}{\pi^{2}}\int_{\Omega_{N}}\int_{\Omega_{N}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(% z^{\prime})\frac{1}{1-tR_{2}(z)}S_{2,2}(z,z^{\prime})\,\mathrm{d}z\mathrm{d}z^% {\prime}\\ &\displaystyle\quad+\frac{2}{\pi^{2}}\int_{\Omega_{N}}\int_{\Omega_{N}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(% z^{\prime})\frac{1}{1-tR_{2}(z)}S_{2,3}(z,z^{\prime})\,\mathrm{d}z\mathrm{d}z.% \end{split} (6.54)

By our assumptions (6.2), (6.1), (6.6), the error term in (6.4) is bounded by

 N^{3\xi}(1+|x|)\mathcal{O}\left(\frac{t^{1/2}}{N^{1/2}t_{1}}\right)+N^{3\xi}|x% |(1+|x|)\mathcal{O}\left(\frac{1}{(Nt_{1})^{1/2}}\right)=(1+|x|)\mathcal{O}(N^% {\omega_{0}/2-\omega_{1}+3\xi})+|x|^{2}\mathcal{O}(N^{-\omega_{1}/2+3\xi}).

Integrating (6.4) from x=0 to |x|\leq N^{\omega_{1}/4-\omega_{0}/8-3\xi} using (6.5), we find:

 \psi(x)=\exp\left(-\frac{x^{2}}{2}V(\varphi_{N})\right)+\mathcal{O}(N^{\omega_% {0}/4-\omega_{1}/2}), (6.55)

which is the assertion of Theorem 6.1.

### 6.5 Computation of T_{1}

###### Proposition 6.6.

We have the estimate:

 \begin{split}\displaystyle T_{1}&\displaystyle=t\tilde{R}_{2}(z)\cdot\mathbb{E% }[e(x)\mathrm{tr}G(\tau+\mathrm{i}\eta)^{\circ}]+\mathcal{O}_{\prec}(N^{-1/2}|% \eta|^{-3/2})\\ &\displaystyle\quad+|x|\mathcal{O}_{\prec}\left(|\eta|^{-1}N^{-1/2}\|\varphi_{% N}^{\prime\prime}\|^{1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\right).\end{split} (6.56)

uniformly for z=\mathcal{D}_{\xi,q}\cup\overline{\mathcal{D}}_{\xi,q}.

We choose k=3 in (6.38) and write:

 \begin{split}\displaystyle T_{1}=&\displaystyle\sum_{j=1}^{N}\frac{\mathbb{E}[% e_{j}^{\circ}\mathbb{E}_{j}[A_{j}^{\circ}(\tau+\mathrm{i}\eta)]]}{\mathbb{E}[A% _{j}(\tau+\mathrm{i}\eta)]^{2}}-\sum_{j=1}^{N}\frac{\mathbb{E}[e_{j}^{\circ}% \mathbb{E}_{j}[(A^{\circ}_{j}(\tau+\mathrm{i}\eta))^{2}]]}{\mathbb{E}[A_{j}(% \tau+\mathrm{i}\eta)]^{3}}+\sum_{j=1}^{N}\frac{1}{\mathbb{E}[A_{j}(\tau+% \mathrm{i}\eta)]^{4}}\mathbb{E}\left[\frac{e_{j}^{\circ}(A^{\circ}_{j})^{3}}{1% +\frac{A^{\circ}_{j}}{\mathbb{E}A_{j}}}\right],\end{split} (6.57)

where we have denoted by \mathbb{E}_{j} integation over the first row of H and have used that e_{j} is independent of this row. The first term on the right of (6.57) will be seen to be the main term in (6.56). To deal with the second term, we compute

 \begin{split}\displaystyle\mathbb{E}_{j}[(A^{\circ}_{j})^{2}]&\displaystyle=% \mathbb{E}_{j}[(-\sqrt{t}w_{jj}+\sum_{kl}^{(j)}h_{jk}G^{(j)}_{kl}h_{lj}-\frac{% t}{N}\mathbb{E}[\mathrm{tr}G^{(j)}])^{2}]\\ &\displaystyle=\frac{t}{N}+\mathbb{E}_{j}[(\sum^{(j)}_{kl}G^{(j)}_{kl}(h_{jk}h% _{jl}-N^{-1}t\delta_{kl}))^{2}]+N^{-2}t^{2}\mathbb{E}_{j}[(\mathrm{tr}G^{(j)}-% \mathbb{E}\mathrm{tr}G^{(j)})^{2}].\end{split} (6.58)

We further compute, using the local law:

 \begin{split}\displaystyle\mathbb{E}_{j}[(\sum^{(j)}_{kl}G^{(j)}_{kl}(h_{jk}h_% {jl}-N^{-1}t\delta_{kl}))^{2}]&\displaystyle=\frac{t^{2}}{N^{2}}\sum^{(j)}_{kl% }G^{(j)}_{kl}G^{(j)}_{kl}=\frac{t^{2}}{N}\partial_{z}m_{N}+\mathcal{O}_{\prec}% (t^{2}N^{-2}|\eta|^{-2}).\end{split} (6.59)

Inserting (6.58), (6.59) into (6.57) and using |e^{\circ}_{j}|\leq 2, \mathbb{E}e^{\circ}_{j}=0, we find:

 \displaystyle T_{1}(\tau,\eta) \displaystyle=\sum_{j=1}^{N}\frac{\mathbb{E}[e_{j}^{\circ}\mathbb{E}_{j}[A_{j}% ^{\circ}(\tau+\mathrm{i}\eta)]]}{\mathbb{E}[A_{j}(\tau+\mathrm{i}\eta)]^{2}} (6.60) \displaystyle+\sum_{j=1}^{N}\frac{1}{|\mathbb{E}[A_{j}]|^{3}}\cdot\mathcal{O}(% t^{2}N^{-2}|\eta|^{-2})+\sum_{j=1}^{N}\frac{1}{|\mathbb{E}[A_{j}]|^{4}}% \mathcal{O}(t^{3/2}N^{-3/2}+t^{3}N^{-2}|\eta|^{-2})). (6.61)

For the last term we have also used (6.28).

Note that

 \displaystyle\mathbb{E}_{j}(A_{j})^{\circ} \displaystyle=\frac{t}{N}\mathrm{tr}G^{(j)}-\frac{t}{N}\mathbb{E}\mathrm{tr}G^% {(j)}=t(m_{N}^{(j)})^{\circ},

and so (6.35) implies

 N\left|\mathbb{E}[\mathbb{E}_{j}[(A_{j})^{\circ}]e_{j}]-\mathbb{E}[(tm_{N})^{% \circ}e_{j}]\right|\leq 2t\mathbb{E}[|(A_{j}^{-1}(1+B_{j}))^{\circ}|]\prec tN^% {-1/2}|\eta|^{-3/2}. (6.62)

It now follows from (6.37) that

 \begin{split}\displaystyle T_{1}&\displaystyle=\sum_{j=1}^{N}\frac{t\mathbb{E}% [e_{j}^{\circ}m_{N}]}{\mathbb{E}[A_{j}(\tau+i\eta)]^{2}}+\sum_{j=1}^{N}\frac{1% }{|\mathbb{E}[A_{j}(\tau+i\eta)]|^{2}}\cdot\mathcal{O}_{\prec}(tN^{-3/2}|\eta|% ^{-3/2}).\end{split}

We now replace e_{j} in (6.60) by e(x). Using that |\mathrm{e}^{\mathrm{i}a}-\mathrm{e}^{\mathrm{i}b}|\leq|a-b|, we find

 \begin{split}&\displaystyle\left|\mathbb{E}[(m_{N})^{\circ}e_{j}]-\mathbb{E}[(% m_{N})^{\circ}e]\right|\varleq C(1+|x|)\mathbb{E}\left[\left|\int_{\mathbb{C}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})N[((m_{N}^{(j)})^{\circ}(z^{% \prime})-(m_{N})^{\circ}(z^{\prime}))]\,\mathrm{d}z^{\prime}\right||(m_{N})^{% \circ}(z)|\right].\end{split} (6.63)

Here z^{\prime}=s+\mathrm{i}\eta^{\prime}.

To evaluate (6.63), we use |(m_{N})^{\circ}|\prec(N|\eta|)^{-1}, together with the following lemma, for which we will also have use in the next section.

###### Lemma 6.7.

We have the estimate,

 \int_{\mathbb{C}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})N[(m_{N}^{(j% )})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^{\prime}))]\,\mathrm{d}z^{\prime}=|x% |\mathcal{O}_{\prec}(\|\varphi_{N}^{\prime\prime}\|_{L^{1}}\|\varphi_{N}^{% \prime}\|_{L^{1}}/N)^{1/2}.

Proof.   Let \epsilon>0 be a parameter to be determined later. Split the integral into two regions, using the real-valuedness of \varphi_{N}:

 \begin{split}\displaystyle\int_{\mathbb{C}}\partial_{\bar{z}}\tilde{\varphi}_{% N}(z^{\prime})N[(m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^{\prime}))% ]\,\mathrm{d}z^{\prime}&\displaystyle=\mathrm{Re}\int_{\mathcal{D}_{\epsilon,q% }}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})N[((m_{N}^{(j)})^{\circ}(z^% {\prime})-(m_{N})^{\circ}(z^{\prime}))]\,\mathrm{d}z^{\prime}\\ &\displaystyle+\mathrm{Re}\int_{\mathcal{D}_{\epsilon,q}^{c}}\partial_{\bar{z}% }\tilde{\varphi}_{N}(z^{\prime})N\left((m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N% })^{\circ}(z^{\prime})\right)\,\mathrm{d}z^{\prime}.\end{split} (6.64)

For the first integral, we simply estimate the real part by the full modulus. Our task is thus to estimate the sum

 \displaystyle N\left|\int_{\mathcal{D}_{\epsilon,q}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})((m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^{% \prime}))\,\mathrm{d}z^{\prime}\right| (6.65) \displaystyle+ \displaystyle N\left|\int_{\mathcal{D}_{\epsilon,q}^{c}}\eta^{\prime}\varphi_{% N}^{\prime\prime}(s)\chi(\eta^{\prime})[\mathrm{Im}\mbox{ }((m_{N}^{(j)})^{% \circ}(z^{\prime})-(m_{N})^{\circ}(z^{\prime}))]\,\mathrm{d}z^{\prime}\right| (6.66) \displaystyle+ \displaystyle N\int_{\mathcal{D}_{\epsilon,q}^{c}}|\varphi_{N}(s)||\chi^{% \prime}(\eta^{\prime})||(m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^{% \prime})|\,\mathrm{d}z^{\prime} (6.67) \displaystyle+ \displaystyle N\int_{\mathcal{D}_{\epsilon,q}^{c}}|\varphi_{N}^{\prime}(s)||% \eta^{\prime}||\chi^{\prime}(\eta^{\prime})||(m_{N}^{(j)})^{\circ}(z^{\prime})% -(m_{N})^{\circ}(z^{\prime})|\,\mathrm{d}z^{\prime}. (6.68)

Below, we will repeatedly use (6.34) and (6.35) to approximate the quantity (m_{N}^{(j)}(z)-m_{N}(z))^{\circ}, resulting in the bound

 |(m_{N}^{(j)}(z)-m_{N}(z))^{\circ}|\prec(N|\eta|)^{-3/2}. (6.69)

Since \{\chi^{\prime}(\eta^{\prime})\neq 0\}\subset\{|\eta^{\prime}|\geq N^{10C_{V}}% -1\}, using (6.35), (6.65) is bounded by

 \left|\int_{\mathcal{D}_{\epsilon,q}}\mathrm{i}\eta^{\prime}\varphi_{N}^{% \prime\prime}(s)N((m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^{\prime}% ))\,\mathrm{d}z^{\prime}\right|+N^{-4C_{V}}(\|\varphi_{N}^{\prime}\|_{L^{1}}+% \|\varphi_{N}^{\prime}\|_{L^{1}}). (6.70)

The error term here is \mathcal{O}(N^{-2}). Introducing a new parameter \epsilon_{2}, we split the \eta^{\prime} integral in the first term in (6.70) into the regions

 \displaystyle\{N^{\epsilon}/N<|\eta^{\prime}|\leq N^{\epsilon_{2}}/N\}, (6.71) \displaystyle\{N^{\epsilon_{2}}/N\leq|\eta^{\prime}|\leq N^{10C_{V}}\}. (6.72)

In the region (6.71), we use (6.69) to find a bound of

 \int_{\{N^{\epsilon}/N<|\eta^{\prime}|\leq N^{\epsilon_{2}}/N\}}|\eta^{\prime}% ||\varphi_{N}^{\prime\prime}(s)|\mathcal{O}(N^{-1/2}|\eta^{\prime}|^{-3/2})\,% \mathrm{d}s\mathrm{d}\eta^{\prime}\leq CN^{\epsilon_{2}/2-1}\|\varphi_{N}^{% \prime\prime}\|_{L^{1}}.

In (6.72), we integrate by parts in s, and combine (6.69) and analyticity, to find that the term (6.65) is bounded by

 \begin{split}&\displaystyle N\int_{N^{-1+{\epsilon_{2}}}<|\eta^{\prime}|

Optimizing \epsilon_{2}, we find that (6.65) is bounded by \mathcal{O}(N^{-1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\|\varphi_{N}^{% \prime\prime}\|_{L^{1}}^{1/2}). For (6.66), we use the assumption (6.2) on the support of \varphi_{N}^{\prime}. The integration is over

 \{0<|\eta^{\prime}|

In the first region, we have by monotonicity – see [55, Lemma 7.19] for details – |\mathrm{Im}\mbox{ }m_{N}^{\circ}(z^{\prime})|,|\mathrm{Im}\mbox{ }(m_{N}^{(j)% })^{\circ}(z^{\prime})|\prec(N\eta^{\prime})^{-1}, so this term is

 \mathcal{O}_{\prec}\left(\int_{0<|\eta^{\prime}|

For the integral over |\eta^{\prime}|>10, we integrate by parts and use \partial_{\eta^{\prime}}\mathrm{Im}\mbox{ }m_{N}=-\partial_{s}\mathrm{Re}m_{N} to find the estimate

 \displaystyle N\left|\int_{|\eta^{\prime}|>10}\varphi_{N}^{\prime}(s)\partial_% {\eta^{\prime}}(\eta^{\prime}\chi(\eta^{\prime}))((m_{N}^{(j)})^{\circ}(z^{% \prime})-(m_{N})^{\circ}(z^{\prime}))\,\mathrm{d}z^{\prime}\right| \displaystyle+ \displaystyle N\left|\int\varphi_{N}^{\prime}(s)10\chi(10)((m_{N}^{(j)})^{% \circ}(s+10\mathrm{i})-(m_{N})^{\circ}(s+10\mathrm{i}))\,\mathrm{d}s\right|.

We use (6.34), (6.35) to find that the expectation of both terms is bounded by \mathcal{O}_{\prec}(N^{-1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}).

Recalling (6.3), the term (6.67) is estimated by

 \displaystyle N\int_{\Omega_{N}^{c}\cap\{N^{C_{V}}-1\leq|y|\leq N^{C_{V}}\}}|% \varphi_{N}(s)|\,\mathrm{d}z^{\prime}\varleq N\frac{N^{10C_{V}}}{N^{20C_{V}}}.

Assuming (without loss of generality) that C_{V}\geq 5, this is \mathcal{O}(N^{-2}). Using |m_{N}^{\circ}(\eta^{\prime})|\prec(N\eta^{\prime})^{-1}, the term (6.68) is \mathcal{O}(N^{-1}\|\varphi_{N}^{\prime}\|_{L^{1}}). Combining all the above, we find that, for any \epsilon>0:

 \begin{split}&\displaystyle\left|\int_{\mathbb{C}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})N[(m_{N}^{(j)})^{\circ}(z^{\prime})-(m_{N})^{\circ}(z^% {\prime}))]\,\mathrm{d}z^{\prime}\right|=\mathcal{O}_{\prec}(N^{-1/2}\|\varphi% _{N}^{\prime\prime}\|_{L^{1}}^{1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}+N^{-% 1+\epsilon}\|\varphi_{N}^{\prime\prime}\|_{L^{1}}).\end{split} (6.74)

The result now follows by optimizing in \epsilon. ∎

Using the previous lemma in (6.63), we have

 \left|\mathbb{E}[(m_{N})^{\circ}e_{j}]-\mathbb{E}[(m_{N})^{\circ}e]\right|=% \mathcal{O}((N|\eta|)^{-1})\|\varphi_{N}^{\prime\prime}\|^{1/2}_{L^{1}}\|% \varphi_{N}^{\prime}\|_{L^{1}}^{1/2}N^{-1/2}.

By (6.37) and (6.27), we can estimate

 \frac{t}{N}\sum_{j=1}^{N}\frac{1}{|\mathbb{E}A_{j}|^{2}}\prec 1. (6.75)

From this we get that, for all \tau+\mathrm{i}\eta\in\Omega_{N},

 \displaystyle T_{1}(\tau,\eta)=\mathbb{E}[e(x)\mathrm{tr}G(\tau+\mathrm{i}\eta% )^{\circ}]\cdot\frac{t}{N}\sum_{j=1}^{N}\frac{1}{\mathbb{E}[A_{j}]^{2}} (6.76) \displaystyle+\frac{t}{N}\sum_{j=1}^{N}\frac{1}{|\mathbb{E}[A_{j}]|^{2}}\left(% \mathcal{O}_{\prec}(N^{-1/2}|\eta|^{-3/2})\right)+\frac{t}{N}\sum_{j=1}^{N}% \frac{1}{|\mathbb{E}[A_{j}]|^{2}}|x|\mathcal{O}_{\prec}\left(|\eta|^{-1}N^{-1/% 2}\|\varphi_{N}^{\prime\prime}\|^{1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\right) \displaystyle=t\tilde{R}_{2}(z)\cdot\mathbb{E}[e(x)\mathrm{tr}G(\tau+\mathrm{i% }\eta)^{\circ}]+\mathcal{O}_{\prec}(N^{-1/2}(|\eta|^{-3/2}+t^{-1/2}))+|x|% \mathcal{O}_{\prec}\left(|\eta|^{-1}N^{-1/2}\|\varphi_{N}^{\prime\prime}\|^{1/% 2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{1/2}\right).

This is the claim of Proposition 6.6. ∎

### 6.6 Computation of T_{2}

We now compute T_{2}. Recall from the definition (6.45) that

 T_{2}(\tau,\eta)=\sum_{j=1}^{N}\mathbb{E}\left[G_{jj}^{\circ}(\tau+\mathrm{i}% \eta)(e-e_{j})\right].

By (6.38), we have:

 \begin{split}\displaystyle\mathbb{E}\left[G_{jj}^{\circ}(\tau+\mathrm{i}\eta)(% e-e_{j})\right]&\displaystyle=\frac{1}{\mathbb{E}[A_{j}]^{2}}\mathbb{E}\left[A% _{j}^{\circ}(\tau+\mathrm{i}\eta)(e-e_{j})\right]\\ &\displaystyle\quad-\frac{1}{\mathbb{E}[A_{j}]^{2}}\mathbb{E}\left[\frac{1}{% \mathbb{E}[A_{j}]+A_{j}^{\circ}}(A_{j}^{\circ}(\tau+\mathrm{i}\eta))^{2}(e-e_{% j})\right].\end{split} (6.77)

Using the expansion

 \displaystyle\exp(iX^{(j)})-\exp(\mathrm{i}X) \displaystyle=\exp(iX^{(j)})\cdot(1-\exp(\mathrm{i}(X-X^{(j)}))) \displaystyle=\exp(\mathrm{i}X^{(j)})(\mathrm{i}(X^{(j)}-X)+O(|X-X^{(j)}|)^{2}),

we have, by Lemma 6.7,

 \begin{split}&\displaystyle e_{j}(x)-e(x)-\frac{\mathrm{i}x}{\pi}e_{j}(x)\int% \partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})(N\cdot[m_{N}^{(j)}-m_{N}](s+% \mathrm{i}\eta^{\prime}))^{\circ}\,\mathrm{d}z^{\prime}\\ \displaystyle=&\displaystyle|x|^{2}\mathcal{O}_{\prec}(\|\varphi_{N}^{\prime% \prime}\|_{L^{1}}\|\varphi_{N}^{\prime}\|_{L^{1}}N^{-1}),\end{split} (6.78)

with overwhelming probability.

Using (6.78) and (6.28) in (6.77), we get the following expression for T_{2}, which holds for \tau+i\eta\in\Omega_{N}:

 \displaystyle T_{2}= \displaystyle-\sum_{j=1}^{N}\frac{1}{\mathbb{E}[A_{j}(\tau+\mathrm{i}\eta)]^{2% }}\frac{\mathrm{i}x}{\pi}\int\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})% \mathbb{E}\left[e_{j}(x)\left(N[m_{N}^{(j)}-m_{N}](s+\mathrm{i}\eta^{\prime})% \right)^{\circ}A_{j}(\tau+\mathrm{i}\eta)^{\circ}\right]\mathrm{d}z^{\prime} (6.79) \displaystyle+\frac{1}{N}\sum_{j=1}^{N}\frac{t|x|^{2}}{|\mathbb{E}A_{j}(\tau+% \mathrm{i}\eta)|^{2}}\cdot\mathcal{O}((N|\eta|)^{-1/2}+t^{-1/2}N^{-1/2})\|% \varphi_{N}^{\prime\prime}\|_{L^{1}}\|\varphi_{N}^{\prime}\|_{L^{1}} (6.80) \displaystyle+\frac{1}{N}\sum_{j=1}^{N}\frac{|x|}{|\mathbb{E}A_{j}(\tau+% \mathrm{i}\eta)|^{3}}\cdot\mathcal{O}(t^{2}N^{-1/2}|\eta|^{-1}+tN^{-1/2})\|% \varphi_{N}^{\prime\prime}\|^{1/2}_{L^{1}}\|\varphi_{N}^{\prime}\|^{1/2}_{L^{1% }}. (6.81)

We now compute the main term in (6.79). We begin by splitting:

 \displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime}% )\mathbb{E}\left[e_{j}(x)\left(N[m_{N}^{(j)}-m_{N}](s+\mathrm{i}\eta^{\prime})% \right)^{\circ}A_{j}(\tau+\mathrm{i}\eta)^{\circ}\right]\mathrm{d}z^{\prime} (6.82) \displaystyle+ \displaystyle\int_{\Omega_{N}^{c}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{% \prime})\mathbb{E}\left[e_{j}(x)\left(N[m_{N}^{(j)}-m_{N}](s+\mathrm{i}\eta^{% \prime})\right)^{\circ}A_{j}(\tau+\mathrm{i}\eta)^{\circ}\right]\mathrm{d}z^{% \prime}. (6.83)

The term (6.83) is estimated in the same way as the second term in (6.64). Together with (6.28), This gives a bound of \mathcal{O}_{\prec}((t(N|\eta|)^{-1/2}+t^{1/2}N^{-1/2})N^{-1+\xi})\|\varphi_{N% }^{\prime\prime}\|_{L^{1}}. We see that the total contribution to T_{2} of the sum over j of (6.83) is bounded by

 \frac{1}{N}\sum_{j=1}^{N}\frac{t|x|}{|\mathbb{E}[A_{j}(\tau+\mathrm{i}\eta)]|^% {2}}(\mathcal{O}((N|\eta|)^{-1/2})+\mathcal{O}(t^{-1/2}N^{-1/2}))N^{\xi}\|% \varphi_{N}^{\prime\prime}\|_{L^{1}}. (6.84)

For the first term (6.82), we use the expansion (6.39). The main terms are

 \begin{split}\displaystyle T_{2,1}&\displaystyle=-\sum_{j=1}^{N}\frac{ix}{\pi}% \int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{% \mathbb{E}[A_{j}(z)]^{2}}\mathbb{E}\left[e_{j}(x)\frac{B_{j}^{\circ}(z^{\prime% })}{\mathbb{E}[A_{j}(z^{\prime})]}A_{j}(z)^{\circ}\right]\mathrm{d}z^{\prime},% \\ \displaystyle T_{2,2}&\displaystyle=\sum_{j=1}^{N}\frac{ix}{\pi}\int_{\Omega_{% N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{\mathbb{E}[A_{j}(% z)]^{2}}\mathbb{E}\left[e_{j}(x)\frac{A_{j}^{\circ}(z^{\prime})(1+\mathbb{E}B_% {j}(z^{\prime}))}{\mathbb{E}[A_{j}(z^{\prime})]^{2}}A_{j}(z)^{\circ}\right]% \mathrm{d}z^{\prime}.\end{split} (6.85)

The remaining terms will be shown to be error terms:

 \displaystyle T_{2,3} \displaystyle=\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{\mathbb{E}[(A_{j}(z% ^{\prime})]^{2}}\mathbb{E}[A^{\circ}_{j}(z^{\prime})B^{\circ}_{j}(z^{\prime})A% _{j}^{\circ}(z)]\mathrm{d}z^{\prime}, (6.86) \displaystyle T_{2,4} \displaystyle=\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{\mathbb{E}[A_{j}(z^% {\prime})]^{2}}\mathbb{E}\left[\frac{(A_{j}^{\circ}(z^{\prime}))^{2}}{A_{j}(z^% {\prime})}(1+B_{j}(z^{\prime}))A_{j}^{\circ}(z)\right]\mathrm{d}z^{\prime}. (6.87)

Collecting the error terms obtained so far and using (6.37), we find

 \begin{split}\displaystyle I_{1}&\displaystyle=\int_{\Omega_{N}}\partial_{\bar% {z}}\tilde{\varphi}_{N}(z)\frac{T_{2}(z)}{1-t\tilde{R}(z)}\,\mathrm{d}z\\ &\displaystyle=\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{% (T_{2,1}(z)+T_{2,2}(z)+T_{2,3}(z)+T_{2,4}(z))}{1-t\tilde{R}(z)}\,\mathrm{d}z+% \int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\Delta_{1,1}(z)% \mathrm{d}z,\\ \end{split} (6.88)

where \Delta_{1,1} is 1/(1-t\tilde{R}(z)) times the difference between T_{2} and the main term (6.79), restricted to the region \Omega_{N}. |\Delta_{1,1}| is bounded by the sum of the errors (6.80), (6.81) and (6.84).

We have:

 \displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\Delta_{% 1,1}(z)\,\mathrm{d}z \displaystyle=\int_{\Omega_{N}}\mathrm{i}\varphi_{N}^{\prime\prime}(\tau)\eta% \chi(\eta)\Delta_{1,1}(z)\,\mathrm{d}z (6.89) \displaystyle+\int_{\Omega_{N}}\mathrm{i}\varphi_{N}(\tau)\chi^{\prime}(\eta)% \Delta_{1,1}(z)\,\mathrm{d}z (6.90) \displaystyle-\int_{\Omega_{N}}\varphi_{N}^{\prime}(\tau)\eta\chi^{\prime}(% \eta)\Delta_{1,1}(z)\,\mathrm{d}z. (6.91)

We first estimate (6.89). After integration by parts in \tau, and using

 |\partial_{z}\Delta_{1,1}(z)|\leq 2|\eta|^{-1}\max_{|w-z|=|\eta|/2}|\Delta_{1,% 1}(w)|,

this is bounded by

 \displaystyle\int_{\{z:N^{-1+\xi}<|\eta|

Split the \eta integral (6.92) into \{|\eta|\leq t\}, \{|\eta|>t\}, and

 \frac{1}{N}\sum_{j=1}^{N}\frac{1}{|\mathbb{E}[A_{j}]|^{2}}\leq C\log N/(\max(t% ,|\eta|)).

This gives the estimate

 |x|\|\varphi_{N}^{\prime}\|_{L^{1}}\|\varphi_{N}^{\prime\prime}\|_{L^{1}}% \mathcal{O}(t^{1/2}N^{-1/2+2\xi}). (6.94)

With t=N^{\omega_{0}}/N and \|\varphi_{N}^{\prime\prime}\|_{L^{1}}\leq N/N^{\omega_{1}}, this is \mathcal{O}(N^{\omega_{0}/2-\omega_{1}+2\xi}), which is O(N^{-c}) if \xi is small enough. By direct computation and (6.75), the term (6.93) is |x|\mathcal{O}(tN^{-1/2}\log N)\|\varphi_{N}^{\prime\prime}\|_{L^{1}}^{1/2}\|% \varphi_{N}^{\prime}\|_{L^{1}}^{3/2}. For the terms (6.90), (6.91), the integrands are supported in the region \{z:N^{10C_{V}}-1<|\mathrm{Im}\mbox{ }z|<N^{10C_{V}}\}. In this region, we use the bound |\mathbb{E}A_{j}(\tau+\mathrm{i}\eta)|\geq c|\eta|, to obtain a bound of the form C\|\varphi_{N}^{\prime}\|_{L^{1}}N^{-2}. The remaining terms T_{2,1}, T_{2,2}, T_{2,3}, T_{2,4} are computed in the following sections.

### 6.7 Computation of T_{2,1}.

We now compute the term T_{2,1} (6.85). Since e_{j}(x) is independent of (h_{ij})_{i=1}^{N}, we first compute

 \mathbb{E}_{j}[A_{j}^{\circ}(\tau+\mathrm{i}\eta)\frac{B_{j}^{\circ}(s+\mathrm% {i}\eta^{\prime})}{\mathbb{E}[A_{j}(s+\mathrm{i}\eta^{\prime})]}]. (6.95)

For simplicity of notation, we will write G(s) for G(s+\mathrm{i}\eta^{\prime}) and G(\tau) for G(\tau+\mathrm{i}\eta). Similar notational simplifications apply to G^{(j)}(s+\mathrm{i}\eta^{\prime}), m_{N}^{(j)}(s+\mathrm{i}\eta^{\prime}), m_{N}^{(j)}(s+\mathrm{i}\eta^{\prime}), etc.

The result of the following computation is:

###### Proposition 6.8.

Uniformly for \tau+\mathrm{i}\eta,s+\mathrm{i}\eta^{\prime}\in\Omega_{N},

 \begin{split}\displaystyle N\mathbb{E}_{j}[A_{j}^{\circ}(\tau+\mathrm{i}\eta)% \frac{B_{j}^{\circ}(s+\mathrm{i}\eta^{\prime})}{\mathbb{E}[A_{j}(s+\mathrm{i}% \eta^{\prime})]}]=&\displaystyle\frac{2t^{2}}{\mathbb{E}[A_{j}(s+\mathrm{i}% \eta^{\prime})]}\cdot\partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},% t}(s)}{\tau-s+\mathrm{i}(\eta-\eta^{\prime})}\\ &\displaystyle+g_{j}(s)\cdot\mathcal{O}_{\prec}(t^{2}N^{-1}|\eta^{\prime}|^{-2% }|\eta|^{-1}).\end{split} (6.96)

Proof.   We first recenter around the conditional expectations \mathbb{E}_{j}A_{j}, \mathbb{E}_{j}B_{j} instead of the full expectations, using the identity

 \displaystyle\mathbb{E}_{j}[(A-\mathbb{E}[A](B-\mathbb{E}[B]]=\mathbb{E}_{j}[(% A-\mathbb{E}_{j}[A])(B-\mathbb{E}_{j}[B])]+(\mathbb{E}_{j}[A]-\mathbb{E}[A])(% \mathbb{E}_{j}[B]-\mathbb{E}[B]).

This produces an error \mathcal{O}(t^{2}N^{-1}|\eta^{\prime}|^{-2}|\eta|^{-1}). We then write

 \displaystyle N\mathbb{E}_{j}[(A_{j}(z)-\mathbb{E}_{j}A_{j}(z))\frac{(B_{j}(z^% {\prime})-\mathbb{E}_{j}B_{j}(z^{\prime}))}{\mathbb{E}[A_{j}(z^{\prime})]}] \displaystyle= \displaystyle N\mathbb{E}_{j}\left(\sum^{(j)}_{i,k}G^{(j)}_{ik}(\tau)(h_{ji}h_% {kj}-N^{-1}t\delta_{ik})\right)\times\left(\sum^{(j)}_{lkm}\frac{1}{\mathbb{E}% [A_{j}(z^{\prime})]}G^{(j)}_{lk}(s)G^{(j)}_{km}(s)(h_{jl}h_{mj}-N^{-1}t\delta_% {lm})\right) \displaystyle= \displaystyle\frac{2t^{2}}{N\mathbb{E}[A_{j}(s+\mathrm{i}\eta^{\prime})]}% \mathrm{tr}(G^{(j)}(\tau)(G^{(j)}(s))^{2}). (6.97)

Now we use

 \displaystyle\frac{1}{N}\mathrm{tr}G^{(j)}(\tau)(G^{(j)}(s))^{2} \displaystyle=\partial_{z}\mathrm{tr}(G^{(j)}(\tau)G^{(j)}(z))|_{z=s+i\eta^{% \prime}}.

We can write this as

 \frac{1}{N}\partial_{s}\frac{\mathrm{tr}G^{(j)}(\tau)-\mathrm{tr}G^{(j)}(s)}{% \tau-s+\mathrm{i}(\eta-\eta^{\prime})}=\partial_{s}\frac{m_{N}^{(j)}(\tau)-m_{% N}^{(j)}(s)}{\tau-s+\mathrm{i}(\eta-\eta^{\prime})}.

Note the identity:

 \partial_{z^{\prime}}\frac{f(z)-f(z^{\prime})}{z-z^{\prime}}=\int_{0}^{1}(1-% \alpha)f^{\prime\prime}(z^{\prime}+\alpha(z-z^{\prime}))\,\mathrm{d}\alpha. (6.98)

If \eta\eta^{\prime}>0 and |\eta-\eta^{\prime}|<\max(|\eta|,|\eta^{\prime}|)/2, we use (6.98) with

 f(z)=m_{N}^{(j)}(z)-m_{\mathrm{fc},t}(z)

to find

 \displaystyle\partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{% \tau-s+\mathrm{i}(\eta-\eta^{\prime})}+\mathcal{O}_{\prec}(\max_{\alpha\in[z,z% ^{\prime}]}\frac{|\alpha\eta+(1-\alpha)\eta^{\prime}|^{-3}}{N})=\partial_{s}% \frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm{i}(\eta-% \eta^{\prime})}+\mathcal{O}_{\prec}(N^{-1}|\eta^{\prime}|^{-2}|\eta|^{-1}).

If |\eta-\eta^{\prime}|>\max(|\eta|,|\eta^{\prime}|)/2, we perform the differentiation

 \frac{-\partial_{s}m_{N}^{(j)}(s)}{(\tau-s)+\mathrm{i}(\eta-\eta^{\prime})}+% \frac{m_{N}^{(j)}(z)-m_{N}^{(j)}(z^{\prime})}{((\tau-s)+\mathrm{i}(\eta-\eta^{% \prime}))^{2}}.

Using the local law, we replace m_{N}^{(j)}(s),m_{N}^{(j)}(\tau) by m_{\mathrm{fc},t}(s),m_{\mathrm{fc},t}(\tau) with an error \mathcal{O}(N^{-1}|\eta^{\prime}|^{-2}|\eta|^{-1}).

If \eta\eta^{\prime}<0, applying the local law again we find

 \partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm% {i}(\eta-\eta^{\prime})}+\mathcal{O}_{\prec}(N^{-1}|\eta^{\prime}|^{-2})\frac{% 1}{|\eta-\eta^{\prime}|}.

Using Proposition 6.8 in the main term of (6.85), and using (6.27) to replace 1/\mathbb{E}[A_{j}], 1/\mathbb{E}[A_{j}]^{2} by g_{j}, g_{j}^{2} we find, for \tau+i\eta\in\Omega_{N}:

 \displaystyle T_{2,1}(z)= \displaystyle-\frac{2\mathrm{i}x}{\pi}\frac{\mathbb{E}[e(x)]}{N}\sum_{j=1}^{N}% t^{2}\int_{\Omega_{N}}g_{j}(z)^{2}g_{j}(z^{\prime})\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})\partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{% fc},t}(s)}{\tau-s+\mathrm{i}(\eta-\eta^{\prime})}\mathrm{d}z^{\prime} (6.99) \displaystyle-\frac{2\mathrm{i}x}{\pi}\sum_{j=1}^{N}\frac{t^{2}}{N}(\mathbb{E}% [e_{j}(x)-e(x)])\int_{\Omega_{N}}g_{j}(z)^{2}g_{j}(z^{\prime})\mathrm{i}\eta^{% \prime}\varphi_{N}^{\prime\prime}(s)\partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-% m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm{i}(\eta-\eta^{\prime})}\mathrm{d}z^{\prime} (6.100) \displaystyle+\frac{1}{N}\sum_{j=1}^{N}\frac{|x|}{|\mathbb{E}[A_{j}(\tau+% \mathrm{i}\eta)]|^{2}}\int_{\Omega_{N}}\frac{1}{\mathbb{E}|A_{j}(s+\mathrm{i}% \eta^{\prime})|}\cdot\mathcal{O}_{\prec}(t^{2}N^{-1}|\eta^{\prime}|^{-2}|\eta|% ^{-1})|\eta^{\prime}||\varphi_{N}^{\prime\prime}(s)|\,\mathrm{d}z^{\prime} (6.101) \displaystyle+\frac{1}{N}\sum_{j=1}^{N}\frac{t^{2}|x|}{|\mathbb{E}[A_{j}(\tau+% \mathrm{i}\eta)]|^{2}}\int_{\Omega_{N}}\frac{1}{\mathbb{E}|A_{j}(s+\mathrm{i}% \eta^{\prime})|}\cdot(\mathcal{O}_{\prec}((N|\eta^{\prime}|)^{-1})+\mathcal{O}% _{\prec}(t^{-1}(N|\eta|)^{-1})) (6.102) \displaystyle\quad\times|\eta^{\prime}||\varphi_{N}^{\prime\prime}(s)|\left|% \partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm% {i}(\eta-\eta^{\prime})}\right|\,\mathrm{d}z^{\prime} (6.103)

Note that above, we have omitted the terms with support in the region \{\chi(\eta^{\prime})\neq 0\}, as they are smaller than the terms displayed.

For the term (6.101), we use (6.75) and (6.37) to find an estimate

 |x|\mathcal{O}_{\prec}(\log N(N|\eta|)^{-1})\|\varphi_{N}^{\prime\prime}\|_{L^% {1}}. (6.104)

To deal with the remaining terms, we use the following estimates:

###### Proposition 6.9.

If \eta,\eta^{\prime}\in\Omega_{N} and \eta\eta^{\prime}>0, then

 \partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm% {i}(\eta-\eta^{\prime})}=\mathcal{O}(|\eta|^{-1}|\eta^{\prime}|^{-1}), (6.105)

If \eta\eta^{\prime}<0, then

 \partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm% {i}(\eta-\eta^{\prime})}=\mathcal{O}(|\eta-\eta^{\prime}|^{-1}|\eta^{\prime}|^% {-1}). (6.106)

Proof.   By the representation (6.98) the left side of (6.105) is

 \int_{0}^{1}\alpha m_{\mathrm{fc},t}^{(2)}(z^{\prime}+\alpha(z-z^{\prime}))\,% \mathrm{d}\alpha. (6.107)

This is bounded by

 \max_{\zeta\in[z,z^{\prime}]}|m^{\prime\prime}_{\mathrm{fc},t}(\zeta)|\leq C|% \eta|^{-1}|\eta^{\prime}|^{-1}.

For (6.106), we simply perform the differentiation:

 \displaystyle\partial_{s}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{% \tau-s+\mathrm{i}(\eta-\eta^{\prime})} \displaystyle= \displaystyle-\frac{\partial_{s}m_{\mathrm{fc},t}(s)}{\tau-s+\mathrm{i}(\eta-% \eta^{\prime})}+\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{(\tau-s+% \mathrm{i}(\eta-\eta^{\prime}))^{2}}=\mathcal{O}(|\eta^{\prime}|^{-1}|\eta-% \eta^{\prime}|^{-1})+\mathcal{O}(|\eta-\eta^{\prime}|^{-2}).

By (6.105), (6.106), and (6.75) the term (6.102) is bounded by |x|O_{\prec}(\log N(N|\eta|)^{-1})\|\varphi_{N}^{\prime\prime}\|_{L^{1}}. For the term (6.100), we use (6.74), (6.105), (6.106), and integrate by parts in s^{\prime} when |\eta^{\prime}|\leq\|\varphi_{N}^{\prime\prime}\|_{L^{1}} to find an error

 |x|(1+|x|)\mathcal{O}(\log N|\eta|^{-1}\|\varphi_{N}^{\prime\prime}\|_{L^{1}}^% {1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{3/2}N^{-1/2})

We have shown the main term in (6.85) is

 -\frac{2\mathrm{i}x}{\pi}\mathbb{E}[e(x)]\int_{\Omega_{N}}S_{2,1}(z,z^{\prime}% )\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\mathrm{d}z^{\prime}+|x|(1+|% x|)\mathcal{O}(\log NN^{-1/2}|\eta|^{-1})\|\varphi_{N}^{\prime\prime}\|^{1/2}_% {L^{1}}\|\varphi_{N}^{\prime}\|^{1/2},

with

 S_{2,1}(z,z^{\prime})=\frac{t^{2}}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{\prime% })\partial_{s}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{% \prime}}. (6.108)

Multiplying T_{2,1}(z)/(1-t\tilde{R}(z)) by \partial_{\bar{z}}\tilde{\varphi}_{N}(z), and integrating we have:

 \displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{T_% {2,1}(z)}{1-t\tilde{R}_{2}(z)}\mathrm{d}z \displaystyle=-2\mathrm{i}x\frac{\mathbb{E}[e(x)]}{\pi}\int_{\Omega_{N}}\frac{% 1}{1-t\tilde{R}_{2}(z)}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\int_{\Omega_{N% }}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})S_{2,1}(z,z^{\prime})\,% \mathrm{d}z^{\prime}\mathrm{d}z \displaystyle+\int_{\Omega_{N}}\frac{1}{1-t\tilde{R}_{2}(z)}\mathrm{i}\eta\chi% (\eta)\varphi^{\prime\prime}_{N}(\tau)\Delta_{2,1}(z)\,\mathrm{d}z, (6.109)

where

 \Delta_{2,1}(z):=T_{2,1}+\frac{\mathrm{i}\mathbb{E}[e(x)]}{\pi}\int\partial_{% \bar{z}}\tilde{\varphi}_{N}(z^{\prime})S_{2,1}(z,z^{\prime})\,\mathrm{d}z^{\prime}

is analytic in \mathrm{Im}\mbox{ }z>0 and \mathrm{Im}\mbox{ }z<0 and

 \frac{\Delta_{2,1}(z)}{1-\tilde{R}_{2}(z)}=\mathcal{O}(|\eta|^{-1}\log N/N^{1/% 2})(|x|(1+|x|)\|\varphi^{\prime\prime}_{N}\|_{L^{1}}^{1/2}\|\varphi_{N}^{% \prime}\|_{L^{1}}^{3/2}.

Integrating by parts in \tau in the integral (6.109) and using

 \partial_{\tau}\frac{\Delta_{2,1}(z)}{1-t\tilde{R}_{2}(z)}=\mathcal{O}(|\eta|^% {-2}\log N/N^{1/2})(|x|(1+|x|))\|\varphi^{\prime\prime}_{N}\|^{1/2}_{L^{1}}\|% \varphi_{N}^{\prime}\|_{L^{1}}^{3/2},

we find

 \displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{T_% {2,1}(z)}{1-t\tilde{R}_{2}(z)}\mathrm{d}z \displaystyle=-\frac{2\mathrm{i}x}{\pi}\mathbb{E}[e(x)]\int_{\Omega_{N}}\frac{% 1}{1-t\tilde{R}_{2}(z)}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\int_{\Omega_{N% }}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})S_{2,1}(z,z^{\prime})\,% \mathrm{d}z^{\prime}\mathrm{d}z \displaystyle+\mathcal{O}((\log N)^{2}/N^{1/2})|x|(1+|x|)\|\varphi^{\prime% \prime}_{N}\|^{1/2}_{L^{1}}\|\varphi_{N}^{\prime}\|^{5/2}_{L^{1}}. (6.110)

### 6.8 Computation T_{2,2}, T_{2,3}, T_{2,4}

The computation of T_{2,2} is almost identical (but simpler) to that in Proposition 6.8.

###### Proposition 6.10.

There are constants for t+\mathrm{i}\eta,s+\mathrm{i}\eta^{\prime}\in\Omega_{N},

 \begin{split}&\displaystyle N\mathbb{E}_{j}[A^{\circ}_{j}(\tau+\mathrm{i}\eta)% (1+\mathbb{E}B_{j}(z^{\prime}))A_{j}^{\circ}(s+\mathrm{i}\eta^{\prime})]\\ \displaystyle=&\displaystyle\,(1+t\partial_{z}m_{\mathrm{fc},t}(z^{\prime}))% \cdot\left(2t^{2}\frac{m_{\mathrm{fc},t}(\tau)-m_{\mathrm{fc},t}(s)}{\tau-s+% \mathrm{i}(\eta-\eta^{\prime})}\cdot(1+\mathcal{O}(tN^{-1}|\eta^{\prime}|^{-2}% ))+t\right)\\ \displaystyle+&\displaystyle\mathcal{O}_{\prec}(t^{2}N^{-1}|\eta|^{-1}|\eta^{% \prime}|^{-1}).\end{split} (6.111)

We have shown that

 \displaystyle T_{2,2}(z) \displaystyle=\frac{2\mathrm{i}x}{\pi}\mathbb{E}[e(x)]\int_{\Omega_{N}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\int(S_{2,2}(z,z^{\prime})+S_% {2,3}(z,z^{\prime}))\,\mathrm{d}z^{\prime}\mathrm{d}z+\Delta_{2,2}(z)

where

 \displaystyle S_{2,2}(z,z^{\prime}) \displaystyle=\frac{t^{2}}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{\prime})^{2}(1% +t\partial_{z}m_{\mathrm{fc},t}(z^{\prime}))\frac{m_{\mathrm{fc},t}(z)-m_{% \mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}, \displaystyle S_{2,3}(z,z^{\prime}) \displaystyle=\frac{t}{N}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{\prime})^{2}(1+t% \partial_{z}m_{\mathrm{fc},t}(z^{\prime})).

\Delta_{2,2}(z) is analytic in \mathrm{Im}\mbox{ }z\neq 0 and

 \displaystyle|\Delta_{2,2}(z)|=\frac{|x|}{N}\sum_{j=1}^{N}\frac{1}{|\mathbb{E}% [A_{j}(z)]|^{2}}\int_{\Omega_{N}}\frac{1}{|\mathbb{E}[A_{j}(z^{\prime})]^{2}|}% \mathcal{O}(t^{2}N^{-1}|\eta|^{-1}|\eta^{\prime}|^{-1})|\eta^{\prime}||\varphi% ^{\prime\prime}(s)|\,\mathrm{d}z^{\prime} \displaystyle+\frac{|x|}{N}\sum_{j=1}^{N}\frac{1}{|\mathbb{E}[A_{j}(z)]|^{2}}% \int_{\Omega_{N}}\frac{1}{|\mathbb{E}[A_{j}(z^{\prime})]^{2}|}(t^{2}\min(|\eta% |^{-1},|\eta^{\prime}|^{-1})+t)\mathcal{O}((tN)^{-1}(|\eta|^{-1}+|\eta^{\prime% }|^{-1}))|\eta^{\prime}||\varphi^{\prime\prime}(s)|\,\mathrm{d}z^{\prime} \displaystyle+\left|\frac{x}{N}\sum_{j=1}^{N}\mathbb{E}[e(x)-e_{j}(x)]g_{j}(z)% ^{2}\int_{\Omega_{N}}g_{j}(z^{\prime})^{2}\eta^{\prime}\varphi_{N}^{\prime% \prime}(s)\left(2it^{2}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}% )}{z-z^{\prime}}+t\right)\,\mathrm{d}z^{\prime}\right| \displaystyle=|x|(1+|x|)\mathcal{O}(|\eta|^{-1}\log NN^{-1/2}\|\varphi_{N}^{% \prime\prime}\|_{L^{1}}^{1/2}\|\varphi_{N}^{\prime}\|_{L^{1}}^{3/2}).

We have used (6.75) and (6.37).

Using the derivative bound

 \partial_{\tau}\frac{\Delta_{2,2}(z)}{1-t\tilde{R}_{2}(z)}=|x|(1+|x|)\mathcal{% O}(|\eta|^{-2}\log N/N^{1/2})\|\varphi^{\prime\prime}_{N}\|^{1/2}_{L^{1}}\|% \varphi_{N}^{\prime}\|_{L^{1}}^{3/2},

we have

 \begin{split}\displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{% N}(z)\frac{\Delta_{2,2}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z&\displaystyle=% \int_{\Omega_{N}}\mathrm{i}\varphi_{N}^{\prime}(\tau)\eta\chi(\eta)\partial_{% \tau}\frac{\Delta_{2,2}(z)}{1-t\tilde{R}_{2}(z)}\mathrm{d}z\\ &\displaystyle=|x|(1+|x|)\mathcal{O}((\log N)^{2}/N^{1/2})\|\varphi^{\prime}_{% N}\|_{L^{1}}^{5/2}\|\varphi_{N}^{\prime\prime}\|_{L^{1}}^{1/2}.\end{split} (6.112)

For T_{2,3}, we use (6.28), (6.29), to estimate the integrand by

 \frac{1}{N}\sum_{j=1}^{N}\frac{|x|}{|\mathbb{E}[A_{j}(z)]|^{2}}\frac{t}{|% \mathbb{E}[A_{j}(z^{\prime})]|^{2}}\mathcal{O}\left(t^{2}|\eta^{\prime}|^{-1/2% }|\eta|^{-1/2}+t^{3/2}|\eta^{\prime}|^{-1/2}+t^{3/2}|\eta|^{-1/2}+t\right)N^{-% 1/2}|\eta^{\prime}|^{-3/2}. (6.113)

The terms to estimate are

 \displaystyle T_{2,3}(z) \displaystyle=\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\mathrm{i}\eta^{\prime}\chi(\eta^{\prime})\varphi_{N}^{\prime\prime}(s)\frac{% 1}{\mathbb{E}[(A_{j}(z^{\prime})]^{2}}\mathbb{E}[A^{\circ}_{j}(z^{\prime})B^{% \circ}_{j}(z^{\prime})A_{j}^{\circ}(z)]\mathrm{d}z^{\prime} (6.114) \displaystyle+\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\mathrm{i}\varphi_{N}(s)\chi^{\prime}(\eta^{\prime})\frac{1}{\mathbb{E}[(A_{j% }(z^{\prime})]^{2}}\mathbb{E}[A^{\circ}_{j}(z^{\prime})B^{\circ}_{j}(z^{\prime% })A_{j}^{\circ}(z)]\mathrm{d}z^{\prime} (6.115) \displaystyle-\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\eta^{\prime}\varphi_{N}^{\prime}(s)\chi^{\prime}(\eta^{\prime})\frac{1}{% \mathbb{E}[(A_{j}(z^{\prime})]^{2}}\mathbb{E}[A^{\circ}_{j}(z^{\prime})B^{% \circ}_{j}(z^{\prime})A_{j}^{\circ}(z)]\mathrm{d}z^{\prime}. (6.116)

Use \{z^{\prime}:|\mathrm{Im}\mbox{ }z^{\prime}|\geq N^{10C_{V}}-1\} on the support of the integrands, and (6.113) to estimate the terms (6.115), (6.116) by

 N^{-10C_{V}}\sum_{j=1}^{N}\frac{|x|}{|\mathbb{E}[A_{j}(z)]|^{2}}|\eta|^{-1/2}(% \|\varphi_{N}\|_{L^{1}}+\|\varphi_{N}^{\prime}\|_{L^{1}}). (6.117)

Inserting the bound (6.113) into (6.114), using |\mathbb{E}[A_{j}(z)]|\geq ct, for |\eta|\leq t and |\mathbb{E}[A_{j}(z)]|\geq|\eta| for |\eta|\geq t, we find

 \frac{1}{N}\sum_{j=1}^{N}\frac{t^{-1}|x|}{|\mathbb{E}{[A_{j}(z)]|^{2}}}N^{-1/2% }\mathcal{O}(t^{2}|\eta|^{-1/2}\log N+t^{3/2}\log N+t^{2}|\eta|^{-1/2}+t^{3/2}% )\|\varphi_{N}^{\prime\prime}\|_{L^{1}}. (6.118)

Integrating (\partial_{\bar{z}}\tilde{\varphi}_{N}(z)T_{2,3}(z))/(1-t\tilde{R}_{2}(z)) over \Omega_{N}, and integrating by parts:

 \displaystyle\int_{\Omega_{N}}\mathrm{i}\eta\chi(\eta)\varphi_{N}^{\prime% \prime}(\tau)\frac{T_{2,3}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z= \displaystyle\int_{\Omega_{N}}\mathrm{i}\partial_{\eta}(\eta\chi(\eta))\varphi% _{N}^{\prime}(\tau)\frac{T_{2,3}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z \displaystyle- \displaystyle N^{-1+\xi}\int\mathrm{i}\chi(\eta)\varphi_{N}^{\prime}(\tau)% \frac{T_{2,3}(\tau+\mathrm{i}N^{-1+\xi})}{1-t\tilde{R}_{2}(\tau+\mathrm{i}N^{-% 1+\xi})}\,\mathrm{d}\tau

Using (6.117), (6.118), these terms are bounded by

 |x|\mathcal{O}(t^{1/2}N^{-1/2}\log N\|\varphi_{N}^{\prime\prime}\|_{L^{1}}\|% \varphi_{N}\|_{L^{1}})=|x|\mathcal{O}(N^{-1/2+(1/2)\omega_{0}-\omega_{1}}\log N% )\|\varphi_{N}^{\prime\prime}\|_{L^{1}}. (6.119)

The contribution to

 \int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{T_{2,3}(z)}{1-t% \tilde{R}_{2}(z)}\,\mathrm{d}z

of the terms involving \chi^{\prime}(\eta)\varphi_{N}(\tau), \chi^{\prime}(\eta)\varphi_{N}(\tau) is easily estimated using the support property of \chi^{\prime}, and found to be

 |x|\mathcal{O}(N^{-2C_{V}})\|\varphi_{N}^{\prime\prime}\|_{L^{1}}.

For T_{2,4}, we again have three terms

 \displaystyle T_{2,4} \displaystyle=\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\frac{\mathrm{i}\varphi_{N}^{\prime\prime}(s)\eta^{\prime}\chi(\eta^{\prime})% }{\mathbb{E}[A_{j}(z^{\prime})]^{2}}\mathbb{E}\left[\frac{(A_{j}^{\circ}(z^{% \prime}))^{2}}{A_{j}(z^{\prime})}(1+B_{j}(z^{\prime}))A_{j}^{\circ}(z)\right]% \mathrm{d}z^{\prime} (6.120) \displaystyle+\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\frac{\mathrm{i}\varphi_{N}(s)\chi^{\prime}(\eta^{\prime})}{\mathbb{E}[A_{j}(% z^{\prime})]^{2}}\mathbb{E}\left[\frac{(A_{j}^{\circ}(z^{\prime}))^{2}}{A_{j}(% z^{\prime})}(1+B_{j}(z^{\prime}))A_{j}^{\circ}(z)\right]\mathrm{d}z^{\prime} (6.121) \displaystyle-\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}\int_{\Omega_{N}% }\frac{\mathrm{i}\varphi_{N}^{\prime}(s)\eta^{\prime}\chi^{\prime}(\eta^{% \prime})}{\mathbb{E}[A_{j}(z^{\prime})]^{2}}\mathbb{E}\left[\frac{(A_{j}^{% \circ}(z^{\prime}))^{2}}{A_{j}(z^{\prime})}(1+B_{j}(z^{\prime}))A_{j}^{\circ}(% z)\right]\mathrm{d}z^{\prime}. (6.122)

We use (6.28), (6.75) to find that the integrand in (6.120)-(6.122) is bounded by

 \displaystyle\frac{|x|}{N}\sum_{j=1}^{N}\frac{t}{\mathbb{E}[|A_{j}(z)]|^{2}}% \frac{1}{|\mathbb{E}[A_{j}(z^{\prime})]|^{2}}\mathbb{E}\frac{1}{|A_{j}(z^{% \prime})|} \displaystyle\times \displaystyle\mathcal{O}\left(t^{2}|\eta^{\prime}|^{-1}(N|\eta|)^{-1/2}+t^{3/2% }|\eta^{\prime}|^{-1}N^{-1/2}+t(N|\eta|)^{-1/2}+t^{1/2}N^{-1/2}\right). (6.123)

For (6.120), we first integrate by parts to write this term as

 \begin{split}&\displaystyle\sum_{j=1}^{N}\frac{x}{\mathbb{E}[A_{j}(z)]^{2}}% \int_{\Omega_{N}}\frac{\mathrm{i}\varphi_{N}^{\prime}(s)\partial_{\eta^{\prime% }}(\eta^{\prime}\chi(\eta^{\prime}))}{\mathbb{E}[A_{j}(z^{\prime})]^{2}}% \mathbb{E}\left[\frac{(A_{j}^{\circ}(z^{\prime}))^{2}}{A_{j}(z^{\prime})}(1+B_% {j}(z^{\prime}))A_{j}^{\circ}(z)\right]\mathrm{d}z^{\prime}\\ &\displaystyle-\sum_{j=1}^{N}\frac{xN^{-1+\xi}}{\mathbb{E}[A_{j}(z)]^{2}}\int_% {\Omega_{N}}\frac{\mathrm{i}\varphi_{N}^{\prime}(s)\chi(\eta^{\prime})}{% \mathbb{E}[A_{j}(s+\mathrm{i}N^{-1+\xi})]^{2}}\mathbb{E}\left[\frac{(A_{j}^{% \circ}(s+\mathrm{i}N^{-1+\xi}))^{2}}{A_{j}(s+\mathrm{i}N^{-1+\xi})}(1+B_{j}(s+% \mathrm{i}N^{-1+\xi}))A_{j}^{\circ}(z)\right]\mathrm{d}s.\end{split} (6.124)

Using (6.123) in (6.124), together with the estimate (6.37) when |\eta^{\prime}|\leq t and |A_{j}(z^{\prime})|\geq c|\eta^{\prime}| when |\eta^{\prime}|\geq t, (6.120) is bounded by

 \frac{1}{N}\sum_{j=1}^{N}\frac{|x|}{|\mathbb{E}[A_{j}(z)]|^{2}}\mathcal{O}(% \log N(N|\eta|)^{-1/2}+t^{-1/2}N^{-1/2}\log N+t^{-1/2}N^{-1/2})\|\varphi_{N}^{% \prime}\|_{L^{1}} (6.125)

Using (6.123) again and \{\eta:\chi^{\prime}(\eta)\neq 0\}\subset\{|\eta|\geq N^{C_{V}}-1\}, the terms (6.121) and (6.122) are estimated by

 |x|\mathcal{O}(N^{-2C_{V}}\|\varphi_{N}^{\prime\prime}\|_{L^{1}}).

Using the bound (6.125), we now conclude as in the case of T_{2,3}, by integrating by parts in \tau:

 \begin{split}&\displaystyle\int_{\Omega_{N}}\partial_{\bar{z}}\tilde{\varphi}_% {N}(z)\frac{T_{2,4}(z)}{1-t\tilde{R}_{2}(z)}\,\mathrm{d}z\\ \displaystyle=&\displaystyle\frac{|x|}{N}\sum_{j=1}\int_{\Omega_{N}}\partial_{% \eta}(\eta\chi(\eta))\varphi_{N}^{\prime}(\tau)\frac{|T_{2,4}(z)|}{|1-t\tilde{% R}_{2}(z)|}\,\mathrm{d}z+\mathcal{O}(N^{-2})\\ \displaystyle=&\displaystyle|x|\mathcal{O}((\log N)N^{-1/2}t^{-1/2})\|\varphi_% {N}^{\prime}\|_{L^{1}}^{2}.\end{split} (6.126)

Collecting the error terms (6.94), (6.110), (6.112), (6.119), (6.126), we obtain

This ends the proof of Proposition 6.5. ∎

### 6.9 Variance term

In this section, we give an asymptotic approximation of the expression V(\varphi_{N}) defined in (6.54). This quantity represents the variance of the limiting random variable for the linear statistics of \varphi_{N}. The result is as follows

###### Proposition 6.11.

Recall the definition of V(\varphi_{N}) in (6.54). Then

 V(\varphi_{N})=-\frac{1}{\pi^{2}}\int_{-Ct}^{Ct}\varphi_{N}(\tau)(H\varphi_{N}% ^{\prime})(\tau)\,\mathrm{d}\tau+\mathcal{O}(1). (6.129)

Here, Hf denotes the Hilbert transform:

 (Hf)(x)=\lim_{\epsilon\rightarrow 0}\int f(y)\mathrm{Re}\frac{1}{(x-y)+\mathrm% {i}\epsilon}\,\mathrm{d}y. (6.130)

In particular, for

 \varphi_{N}(x)=\int_{0}^{x}\chi(y/(t_{1}N^{\alpha}))p_{t_{1}}(0,y)\mathrm{d}y,

we have

 V(\varphi_{N})\geq c\log(t/t_{1})\cdot(1+o(1)).

Moreover, if

 \mathrm{supp}\varphi_{N}\subset(-N^{r}t_{1},N^{r}t_{1}),

then

 V(\varphi_{N})=-\frac{1}{\pi^{2}}\int\varphi_{N}(\tau)(H\varphi^{\prime}_{N})(% \tau)\,\mathrm{d}\tau+\mathcal{O}(N^{\omega_{0}/2-\omega_{1}+\xi})

for any \xi>0.

We begin by reducing the domain of integration. Define

 \displaystyle\Omega_{N}^{*}=\{z=E+\mathrm{i}\eta:E\in\mathcal{I}_{q},\,N^{-1+% \xi}<|\eta|\leq N^{10C_{V}}\}.

Note that

 \{\partial_{\bar{z}}\tilde{\varphi}_{N}\neq 0\}\cap(\Omega_{N}\setminus\Omega_% {N}^{*})\subset\{N^{10C_{V}}-1<|\eta|

If either z or z^{\prime} lies in the latter region, then

 \begin{split}\displaystyle\frac{1}{1-tR_{2}(z)}S_{2,1}(z,z^{\prime})&% \displaystyle=\frac{t^{2}}{N(1-tR_{2}(z))}\sum_{j=1}^{N}g_{j}(z)^{2}g_{j}(z^{% \prime})\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}}=\mathcal{O}_{\prec}(t^{2}N^{-4C_{V}}).\end{split} (6.132)

Similarly ,

 \frac{1}{1-tR_{2}(z)}S_{2,2}(z,z^{\prime})=\mathcal{O}(t^{2}N^{-4C_{V}}). (6.133)

With (6.132), (6.133), it is easy to show that the domain of integration \Omega_{N}\times\Omega_{N} in (6.127), (6.128) can be replaced by \Omega_{N}^{*}\times\Omega_{N}^{*} with an error \mathcal{O}(N^{-2}).

Next, we have the following:

###### Proposition 6.12.
 \frac{1}{N}\sum_{j=1}^{N}\frac{g_{j}(z)^{2}}{1-tR_{2}(z)}g_{j}(z^{\prime})=% \partial_{z}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{% \prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))}. (6.134)

Proof.   By [55, Eqn. (7.24)]

By partial fractions, we have

 \displaystyle\frac{1}{N}\sum_{j=1}^{N}\frac{g_{j}(z)^{2}}{1-tR_{2}(z)}g_{j}(z^% {\prime})=\partial_{z}\frac{1}{N}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})=\frac% {1}{N}\partial_{z}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-% z^{\prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))}.

The integrals appearing in the definition of V(\varphi_{N}) are

 \displaystyle I_{1,1} \displaystyle:=\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}\partial_{\bar{z}}% \tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1% }{1-tR_{2}(z)}S_{2,1}(z,z^{\prime})\,\mathrm{d}z\mathrm{d}z^{\prime}, (6.136) \displaystyle I_{1,2} \displaystyle:=\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}\partial_{\bar{z}}% \tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1% }{1-tR_{2}(z)}S_{2,2}(z,z^{\prime})\,\mathrm{d}z\mathrm{d}z^{\prime} (6.137) \displaystyle I_{1,3} \displaystyle:=\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}\partial_{\bar{z}}% \tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1% }{1-tR_{2}(z)}S_{2,3}(z,z^{\prime})\,\mathrm{d}z\mathrm{d}z. (6.138)

By Proposition 6.12,

 I_{1,1}=\frac{t^{2}}{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}\partial_{% \bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime}% )\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})\partial_{s}\frac{m_{% \mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}\,\mathrm{d}z% \mathrm{d}z^{\prime}. (6.139)

Similarly, we have:

 \displaystyle\frac{S_{2,2}(z,z^{\prime})}{1-tR_{2}(z)} \displaystyle=\frac{t^{2}}{N}\partial_{z}\partial_{z^{\prime}}\sum_{j=1}g_{j}(% z)g_{j}(z^{\prime})\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z% -z^{\prime}}, \displaystyle\frac{S_{2,3}(z,z^{\prime})}{1-tR_{2}(z)} \displaystyle=\frac{t}{N}\partial_{z}\partial_{z^{\prime}}\sum_{j=1}g_{j}(z)g_% {j}(z^{\prime}).

Integrating by parts in s=\mathrm{Re}z^{\prime}. The boundary term is only non-zero in the region (6.131), where we can use (6.132).

 \displaystyle I_{1,2}= \displaystyle-\frac{t^{2}}{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{s}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})% \frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}\,% \mathrm{d}z\mathrm{d}z^{\prime} (6.140) \displaystyle-\frac{t^{2}}{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(% z^{\prime})\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})\partial_{s}% \frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}\,% \mathrm{d}z\mathrm{d}z^{\prime} (6.141)

Note that the second integral (6.141) is equal to I_{1,1}.

We begin by computing the z integral in (6.140). The integrand is \partial_{\bar{z}}\tilde{\varphi}_{N}(z) multiplying a function analytic in each of \{\mathrm{Im}\mbox{ }z>0\} and \{\mathrm{Im}\mbox{ }z<0\}. Let \Omega\subset\mathbb{C} be a domain. For F a C^{1}(\Omega) function, Green’s theorem in complex notation is

 \int_{\Omega}\bar{\partial}_{z}F(z)\,\mathrm{d}z=-\frac{\mathrm{i}}{2}\int_{% \partial\Omega}F(z)\,\mathrm{d}z. (6.142)

We split the integral (6.140) into the two regions \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z>0\}, \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z<0\} and apply Green’s theorem to each. The first region is a rectangle in the upper half-plane. The integrand in the resulting line integral, \tilde{\varphi}_{N}, is zero on the “top” segment [-qG+\mathrm{i}N^{10C_{V}},qG+\mathrm{i}N^{10C_{V}}].

We label the terms corresponding to three other boundary line integrals by (+) to denote \mathrm{Im}\mbox{ }z>0 and number them according to the corresponding boundary segments as (1) for [-qG+\mathrm{i}N^{-1+\xi},qG+\mathrm{i}N^{-1+\xi}]; (2) for [qG+\mathrm{i}N^{-1+\xi},qG+\mathrm{i}N^{10C_{V}}]; and (3) for [-qG+iN^{10C_{V}},-qG+iN^{10C_{V}}]:

 \displaystyle 2\mathrm{i}\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z>0\}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{t^{2}}{N}\partial_{z}\sum_{j=1}^% {N}g_{j}(z)g_{j}(z^{\prime})\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}}\,\mathrm{d}z \displaystyle= \displaystyle t^{2}\int_{-qG}^{qG}(\varphi_{N}(\tau)+\mathrm{i}N^{-1+\xi}% \varphi_{N}^{\prime}(\tau))\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau+\mathrm% {i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{\prime})}{\tau+\mathrm{i}N^{-1+\xi}-z^{% \prime}+t(m_{\mathrm{fc},t}(\tau+\mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{% \prime}))} (6.143) \displaystyle\quad\times\frac{m_{\mathrm{fc},t}(\tau+\mathrm{i}N^{-1+\xi})-m_{% \mathrm{fc},t}(z^{\prime})}{\tau+\mathrm{i}N^{-1+\xi}-z^{\prime}}\,\mathrm{d}\tau \displaystyle+ \displaystyle t^{2}\varphi_{N}(qG)\int_{N^{-1+\xi}}^{N^{10C_{V}}+1}\chi(\eta)% \partial_{\tau}\frac{m_{\mathrm{fc},t}(qG+\mathrm{i}\eta)-m_{\mathrm{fc},t}(z^% {\prime})}{qG+\mathrm{i}\eta-z^{\prime}+t(m_{\mathrm{fc},t}(qG+\mathrm{i}\eta)% -m_{\mathrm{fc},t}(z^{\prime}))}\frac{m_{\mathrm{fc},t}(qE+\mathrm{i}\eta)-m_{% \mathrm{fc},t}(z^{\prime})}{qG+\mathrm{i}\eta-z^{\prime}}\,\mathrm{d}\eta (6.144) \displaystyle+ \displaystyle t^{2}\varphi_{N}(-qG)\int_{N^{10C_{V}}+1}^{N^{-1+\xi}}\chi(\eta)% \partial_{\tau}\frac{m_{\mathrm{fc},t}(-qG+\mathrm{i}\eta)-m_{\mathrm{fc},t}(z% ^{\prime})}{-qG+\mathrm{i}\eta-z^{\prime}+t(m_{\mathrm{fc},t}(-qG+\mathrm{i}% \eta)-m_{\mathrm{fc},t}(z^{\prime}))} \displaystyle\times\frac{m_{\mathrm{fc},t}(-qG+\mathrm{i}\eta)-m_{\mathrm{fc},% t}(z^{\prime})}{-qG+\mathrm{i}\eta-z^{\prime}}\,\mathrm{d}\eta (6.145) \displaystyle:=I_{1,+,1}+I_{1,+,2}+I_{1,+,3}. (6.146)

Similarly, the second region \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z<0\} is labelled by (-) in indices. The sides are labelled in counter-clockwise orientation as (1), [qG-\mathrm{i}N^{-1+\xi},-qG-\mathrm{i}N^{-1+\xi}]; (2), [-qG-\mathrm{i}N^{-1+\xi},-qG-\mathrm{i}N^{10C_{V}}]; (3), [qG-\mathrm{i}N^{10C_{V}},qG-\mathrm{i}N^{-1+\xi}]. Applying Green’s theorem to (6.140) over this region:

 \displaystyle 2\mathrm{i}\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z<0\}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\frac{t^{2}}{N}\partial_{z}\sum_{j=1}^% {N}g_{j}(z)g_{j}(z^{\prime})\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}}\,\mathrm{d}z \displaystyle= \displaystyle t^{2}\int_{qG}^{-qG}(\varphi_{N}(\tau)-\mathrm{i}N^{-1+\xi}% \varphi_{N}^{\prime}(\tau))\partial_{z}\frac{m_{\mathrm{fc},t}(\tau-\mathrm{i}% N^{-1+\xi})-m_{\mathrm{fc},t}(z^{\prime})}{\tau-z^{\prime}+t(m_{\mathrm{fc},t}% (\tau-\mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{\prime}))} (6.147) \displaystyle\quad\times\frac{m_{\mathrm{fc},t}(\tau-\mathrm{i}N^{-1+\xi})-m_{% \mathrm{fc},t}(z^{\prime})}{\tau-\mathrm{i}N^{-1+\xi}-z^{\prime}}\,\mathrm{d}x (6.148) \displaystyle+ \displaystyle t^{2}\varphi_{N}(-qG)\int_{-N^{-1+\xi}}^{-N^{10C_{V}}-1}\chi(% \eta)\partial_{z}\frac{m_{\mathrm{fc},t}(-qG+\mathrm{i}\eta)-m_{\mathrm{fc},t}% (z^{\prime})}{-qE+i\eta-z^{\prime}+t(m_{\mathrm{fc},t}(-qG+\mathrm{i}\eta)-m_{% \mathrm{fc},t}(z^{\prime}))} \displaystyle\times\frac{m_{\mathrm{fc},t}(-qG+\mathrm{i}y)-m_{\mathrm{fc},t}(% z^{\prime})}{-qG+\mathrm{i}\eta-z^{\prime}}\,\mathrm{d}\eta (6.149) \displaystyle+ \displaystyle t^{2}\varphi_{N}(qG)\int_{-N^{10C_{V}}-1}^{-N^{-1+\xi}}\chi(\eta% )\partial_{z}\frac{m_{\mathrm{fc},t}(qG+\mathrm{i}\eta)-m_{\mathrm{fc},t}(z^{% \prime})}{qE+i\eta-z^{\prime}+t(m_{\mathrm{fc},t}(qG+\mathrm{i}y)-m_{\mathrm{% fc},t}(z^{\prime}))}\frac{m_{\mathrm{fc},t}(qE+\mathrm{i}\eta)-m_{\mathrm{fc},% t}(z^{\prime})}{qG+\mathrm{i}\eta-z^{\prime}}\,\mathrm{d}\eta (6.150) \displaystyle:=I_{1,-,1}+I_{1,-,2}+I_{1,-,3}.

We now insert -(\mathrm{i}/2)I_{1,\pm,k}(z^{\prime}), k=1,2,3, into the integral (6.140), and apply Green’s theorem in each of the regions \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{\prime}>0\} and \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{\prime}<0\}. We label the oriented sides of that region as previously:

 \displaystyle-(\mathrm{i}/2)\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{% \prime}>0\}}\partial_{s}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})I_{1,% \pm,k}(z^{\prime})\,\mathrm{d}z^{\prime} \displaystyle=\frac{1}{4}\int_{-qG}^{qG}(\varphi_{N}^{\prime}(s)+\mathrm{i}N^{% -1+\xi}\varphi_{N}^{\prime\prime}(s))I_{1,\pm,k}(s+\mathrm{i}N^{-1+\xi})\,% \mathrm{d}s (6.151) \displaystyle+\frac{1}{4}\int_{N^{-1+\xi}}^{N^{10C_{V}}}\chi(\eta^{\prime})(% \varphi_{N}^{\prime}(qG)+\mathrm{i}\eta^{\prime}\varphi^{\prime\prime}_{N}(qG)% )I_{1,\pm,k}(qG+\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{\prime} (6.152) \displaystyle+\frac{1}{4}\int_{N^{10C_{V}}}^{N^{-1+\xi}}\chi(\eta^{\prime})(% \varphi_{N}^{\prime}(-qG)+\mathrm{i}\eta^{\prime}\varphi^{\prime\prime}_{N}(-% qG))I_{1,\pm,k}(qG+\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{\prime} (6.153)

By the support condition (6.2), the terms (6.152) and (6.153) are 0 for any k and choice of \pm. We denote the remaining term (6.151) by I_{1,\pm,k,+}. Similarly, applying Green’s theorem to \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{\prime}<0\}:

 \displaystyle-(\mathrm{i}/2)\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{% \prime}>0\}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})I_{1,\pm,j}(z^{% \prime})\,\mathrm{d}z^{\prime} \displaystyle=\frac{1}{4}\int_{qG}^{-qG}(\varphi_{N}^{\prime}(s)-\mathrm{i}N^{% -1+\xi}\varphi_{N}^{\prime\prime}(s))I_{2,\pm,k}(x^{\prime}-\mathrm{i}N^{-1+% \xi})\,\mathrm{d}x^{\prime} (6.154) \displaystyle:=I_{1,\pm,k,-}

To summarize, we have shown so far

 \displaystyle\frac{t^{2}}{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}\partial% _{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{z^{\prime}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z^{\prime})\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})% \frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}\,% \mathrm{d}z\mathrm{d}z=\sum_{k=1}^{3}\sum_{\alpha,\beta\in\{\pm\}}I_{1,\alpha,% k,\beta}.

Only the terms I_{1,\pm,1,\pm} contribute to the variance. This is the content of the following.

###### Proposition 6.13.

Recall the parameter \sigma>0 in the local law, Lemma A.2. For any choice of I_{2,\pm,k,\pm} with k\neq 1, we have

 |I_{1,\pm,k,\pm}|\leq\mathcal{O}(t^{\sigma}\log N(\|\varphi_{N}^{\prime}\|_{L^% {1}}+N^{-1+\xi}\|\varphi_{N}^{\prime\prime}\|_{L^{1}})).

Proof.   By [55, Eqn (7.25)],

 |\partial_{z}m_{\mathrm{fc},t}(z)|\leq Ct^{-1}. (6.155)

Compute the derivative:

 \displaystyle\partial_{z}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))} \displaystyle=-\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{(z-z^% {\prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})))^{2}}(1+t% \partial_{z}m_{\mathrm{fc},t}(z)) \displaystyle+\frac{\partial_{z}m_{\mathrm{fc},t}(z)}{z-z^{\prime}+t(m_{% \mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))}.

Note that t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))=\mathcal{O}_{\prec}(t) for z,z^{\prime}\in\mathcal{D}_{\epsilon,q}. So if |\mathrm{Re}z|\geq qG\geq t^{1/2}N^{\sigma/2} and |\mathrm{Re}z^{\prime}|=\mathcal{O}(t_{1}N^{r}), then

 \partial_{z}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{% \prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))}\prec t^{-1}t^{-% 1/2}N^{-\sigma/2}\leq t^{-3/2+\sigma/2}. (6.156)

Similarly:

 \left|\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}}% \right|\prec\min(t^{-1/2+\sigma/2},|\eta|^{-1}).

By (6.37), and \frac{1}{N}\sum|g_{j}(z)|\leq C\log N [55, Eqn. (7.36)], so

 \left|\frac{1}{N}\sum_{j=1}^{N}g_{j}(z)^{2}(1+t\partial_{z}m_{\mathrm{fc},t}(z% ))g_{j}(z^{\prime})\right|\leq C|\eta|^{-2}\log N.

Combining this with (6.156), we have

 \left|\frac{1}{N}\sum_{j=1}^{N}g_{j}(z)^{2}(1+t\partial_{z}m_{\mathrm{fc},t}(z% ))g_{j}(z^{\prime})\right|\prec\min(|\eta|^{-2}\log N,t^{-3/2+\sigma/2}).

Inserting this into (6.144), (6.145), (6.149), (6.150), for z^{\prime} such that s=\mathrm{Re}z^{\prime}\in\mathrm{supp}\varphi_{N}^{\prime} and k=2,3:

 \displaystyle|I_{1,\pm,k,\pm}| \displaystyle\leq t^{2}\log N(\|\varphi_{N}^{\prime}\|_{L^{1}}+N^{-1+\xi}\|% \varphi_{N}^{\prime\prime}\|_{L^{1}})\int_{N^{-1+\xi}}^{N^{10C_{V}}}\min(t^{-2% +\sigma},|\eta|^{-3})\mathrm{d}\eta \displaystyle\leq t^{\sigma}\log N(\|\varphi_{N}^{\prime}\|_{L^{1}}+N^{-1+\xi}% \|\varphi_{N}^{\prime\prime}\|_{L^{1}}).

For brevity of notation, we let s^{\pm}=s\pm\mathrm{i}N^{-1+\xi}, \tau^{\pm}=\tau\pm\mathrm{i}N^{-1+\xi}. We have so far shown that

 \displaystyle\int\int_{\Omega_{N}\times\Omega_{N}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\partial_{z}% \sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{% fc},t}(z^{\prime})}{z-z^{\prime}}\,\mathrm{d}z^{\prime}\mathrm{d}z (6.157) \displaystyle= \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(% \tau^{+})-m_{\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{% \mathrm{fc},t}(s^{+}))}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{% +})}{\tau-s}\,\mathrm{d}\tau\mathrm{d}s (6.158) \displaystyle+ \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(% \tau^{-})-m_{\mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{% \mathrm{fc},t}(s^{-}))}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{% -})}{\tau-s}\,\mathrm{d}\tau\mathrm{d}s (6.159) \displaystyle- \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(% \tau^{+})-m_{\mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{% \mathrm{fc},t}(s^{-}))}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{% -})}{\tau-s+2iN^{-1+\xi}}\,\mathrm{d}\tau\mathrm{d}s (6.160) \displaystyle- \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(% \tau^{-})-m_{\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{% \mathrm{fc},t}(s^{+}))}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{% +})}{\tau-s-2iN^{-1+\xi}}\,\mathrm{d}\tau\mathrm{d}s (6.161) \displaystyle+ \displaystyle\mathcal{O}(t^{\sigma}\log N(\|\varphi_{N}^{\prime}\|_{L^{1}}+N^{% -1+\xi}\|\varphi_{N}^{\prime\prime}\|_{L^{1}})). (6.162)

The main terms are (6.160), (6.161). These are of order \log N for the functions we are interested in. The other two terms are bounded by a constant:

###### Proposition 6.14.

Let s^{\pm}=s\pm\mathrm{i}N^{-1+\xi}, \tau^{\pm}=\tau\pm\mathrm{i}N^{-1+\xi}. There is a constant C such that

 \displaystyle\left|t^{2}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(\tau% )\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+% })-m_{\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{% fc},t}(s^{+}))}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+})}{% \tau-s}\,\mathrm{d}\tau\mathrm{d}s\right| \displaystyle\leq C (6.163) \displaystyle\left|t^{2}\int_{-qG}^{qG}\int_{-qG}^{qG}\tilde{\varphi}_{N}(\tau% )\tilde{\varphi}_{N}^{\prime}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-% })-m_{\mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{% fc},t}(s^{-}))}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{-})}{% \tau-s}\,\mathrm{d}\tau\mathrm{d}s\right| \displaystyle\leq C. (6.164)

Proof.   First note the estimate

 \left|\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+}% )}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+}))}\right|\leq Ct% ^{-2}, (6.165)

for |\tau-s|\leq Ct, which follows from the alternate representation

 \frac{1}{N}\sum_{j=1}^{N}g_{j}(s)^{2}(1+t\partial_{s}m_{\mathrm{fc},t}(s))g_{j% }(\tau), (6.166)

(6.155), (6.37), and the bound [55, Eqn (7.24)]

 \frac{1}{N}\sum_{j=1}^{N}\frac{1}{|V_{j}-z-tm_{\mathrm{fc},t}(z)|^{2}}\leq Ct^% {-1}. (6.167)

Define \zeta(z):=z+tm_{\mathrm{fc},t}(z). We begin by noting that (6.135) implies, for z=\tau\pm\mathrm{i}N^{-1+\xi},z^{\prime}=s+\mathrm{i}N^{-1+\xi}\in\Omega_{N}

 \begin{split}\displaystyle\left|\mathrm{Re}(\zeta(z)-\zeta(z^{\prime}))\right|% &\displaystyle=\left|\mathrm{Re}\int_{\tau}^{s}\partial_{x}\zeta(x+\mathrm{i}N% ^{-1+\xi})\,\mathrm{d}x\right|=\left|\int_{\tau}^{s}\mathrm{Re}\frac{1}{1-tR_{% 2}(x+\mathrm{i}N^{-1+\xi})}\,\mathrm{d}x\right|\\ &\displaystyle=\left|\int\int_{\tau}^{z}\frac{\mathrm{Re}(1-tR_{2}(x-\mathrm{i% }N^{-1+\xi})}{|1-tR_{2}(x+\mathrm{i}N^{-1+\xi})|^{2}}\,\mathrm{d}x\right|% \vargeq C|\tau-s|.\end{split} (6.168)

In the second to last step we have used (6.167) as well as the lower bound \mathrm{Re}(1-tR_{2}(z))\vargeq c, (see [55, Lemma 7.2]).

We then estimate the integral in (6.163) as

 \begin{split}&\displaystyle\int_{-qG}^{qG}|\varphi_{N}^{\prime}(s)|\int_{\tau:% |\tau-s|

where M is some constant. In the range \{\tau:|\tau-s|<Mt\}, we use (6.155) in the inner integral, to obtain a bound of constant order. Using that |m_{\mathrm{fc},t}|\leq C in \mathcal{I}_{q}, and (6.168), we have for |\tau-s|\geq Mt:

 \left|\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+}% )}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+}))}\right|\leq C% \frac{t^{-1}}{|s-\tau|}, (6.170)

while

 \frac{|m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(\tau^{-})|}{|\tau-s|}\leq% \frac{C}{|s-\tau|}.

Integrating over \{\tau:|\tau-s|\geq Mt\} then again gives a constant bound for the \tau integral. Since \|\varphi_{N}^{\prime}\|_{L^{1}}\leq C by assumption, we are done. ∎

Summing the two terms (6.160), (6.161), we find a kernel multiplying \tilde{\varphi}_{N}^{\prime}(s)\tilde{\varphi}_{N}(\tau), equal to

 \displaystyle 2t^{2}\mathrm{Re}\,\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{% +})-m_{\mathrm{fc},t}(s^{-})}{\tau-s-2\mathrm{i}N^{-1+\xi}+t(m_{\mathrm{fc},t}% (\tau^{+})-m_{\mathrm{fc},t}(s^{-}))}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{% \mathrm{fc},t}(s^{-})}{\tau-s-2\mathrm{i}N^{-1+\xi}} \displaystyle= \displaystyle 2t^{2}\mathrm{Re}\,\frac{\partial_{\tau}m_{\mathrm{fc},t}(\tau^{% +})(\tau-s+2\mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(\tau^{+})+m_{\mathrm{fc},t% }(s^{-})}{(\tau-s-2\mathrm{i}N^{-1+\xi}+t(m_{\mathrm{fc},t}(\tau^{+})-m_{% \mathrm{fc},t}(s^{-})))^{2}}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t% }(s^{-})}{\tau-s-2\mathrm{i}N^{-1+\xi}}.

Recall:

 \lim_{\epsilon\rightarrow 0}m_{\mathrm{fc},t}(x+\mathrm{i}\epsilon)=(H\rho_{% \mathrm{fc},t})(x)+\mathrm{i}\pi\rho(x),

so

 m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{-})=2\mathrm{i}\pi\rho_{% \mathrm{fc},t}(\tau)+\max_{z}|m_{\mathrm{fc},t}^{\prime}(z)|\cdot\mathcal{O}(|% \tau-s|), (6.171)

so the kernel is

 -2\mathrm{Re}\frac{1}{\tau-s+2\mathrm{i}N^{-1+\xi}}+\max_{z}|m_{\mathrm{fc},t}% ^{\prime}(z)|\frac{t^{2}}{t^{2}+|\tau-s|^{2}}\mathcal{O}\left(\frac{|\tau-s|}{% |\tau-s+2\mathrm{i}N^{-1+\xi}|}\right),

when |\tau-s|\leq Mt, provided M is sufficiently small. Integrating \tilde{\varphi}_{N}(\tau)\tilde{\varphi}_{N}^{\prime}(s) against the error term, using (6.155), and splitting the \tau integral according to |\tau-s|\leq Mt and |\tau-s|>Mt, we find an error term of

 C(\|\varphi_{N}^{\prime}\|_{L^{1}}+N^{-1+\xi}\|\varphi_{N}^{\prime\prime}\|_{L% ^{1}})\|\varphi_{N}\|_{L^{1}}\leq C.

The main term of (6.157) is then

 \begin{split}&\displaystyle\frac{1}{2}\int_{-tM<\tau

For the second step, we have used

 \displaystyle\mathrm{Re}\int_{|\tau-s|\leq N^{10}}\frac{1}{\tau-s+\mathrm{i}2N% ^{-1+\xi}}\,\mathrm{d}\tau=\int_{|\tau-s|\leq N^{10}}\frac{\tau-s}{(\tau-s)^{2% }+4N^{-2+2\xi}}\,\mathrm{d}\tau=0,

to write:

 \displaystyle\mathrm{Re}\int_{-tM<\tau

The difference between the last expression and \int_{-tM<\tau<tM}\varphi_{N}(\tau)(H\varphi_{N}^{\prime})(\tau)\,\mathrm{d}\tau is

 \mathrm{Re}\int_{-tM<\tau

We split the inner integral into |\tau-s|\leq\delta and |\tau-s|>\delta to find the estimate

 t\|\varphi_{N}^{\prime}\|_{C^{\alpha}}\delta^{\alpha}+\frac{N^{-1+\xi}}{\delta% }\|\varphi_{N}^{\prime}\|_{L^{1}}.

Optimizing in \delta, and using \|\varphi_{N}^{\prime}\|_{L^{1}}\leq C we get a bound of

 (t\|\varphi_{N}^{\prime}\|_{C^{\alpha}})^{\frac{1}{1+\alpha}}N^{(-1+\xi)\frac{% \alpha}{1+\alpha}}.

The right side is bounded by

 CN^{\frac{\omega_{0}}{1+\alpha}-\omega_{1}+\frac{\alpha}{1+\alpha}\xi}.

Using condition (6.6), for \alpha>0 small enough, this is \mathcal{O}(N^{-c}).

We now proceed to computing I_{1,1} (6.139). Recall the definition of the constant M introduced in Proposition 6.14. Note first that by the support assumption on \varphi_{N}, \partial_{\bar{z}}\tilde{\varphi}_{N}=0 in \{|E|>Mt\} we can replace the integration domain \Omega_{N}^{*} by:

 \Omega_{N}^{**}=\Omega_{N}^{*}\cap\{E+\mathrm{i}\eta:|E|\leq Mt\}.

We use Green’s theorem (6.142) on \Omega_{N}^{**}\cap\{\mathrm{Im}\mbox{ }z>0\}. The boundary of this region consists of four segments. The function \tilde{\varphi}_{N}(z) is zero on the top segment is [-tM+\mathrm{i}N^{10C_{V}},tM+\mathrm{i}N^{10C_{V}}]. We number the remaining parts of the boundary as: (1) [-tM+\mathrm{i}N^{-1+\xi},tM+\mathrm{i}N^{-1+\xi}]; (2) [-tM+\mathrm{i}N^{10C_{V}},-tM+\mathrm{i}N^{-1+\xi}]; and (3) [tM+\mathrm{i}N^{-1+\xi},tM+\mathrm{i}N^{10C_{V}}]. The result of our application of Green’s theorem to

 \frac{t^{2}}{N}\int_{\Omega_{N}^{**}\cap\{\mathrm{Im}\mbox{ }z>0\}}\partial_{% \bar{z}}\tilde{\varphi}_{N}(z)\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{% \prime})\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}}\,\mathrm{d}z\mathrm{d}z^{\prime} (6.173)

is a sum of line integrals which we label according to the corresponding parts of the boundary. For (1), we have:

 \begin{split}\displaystyle I_{2,+,1}&\displaystyle=t^{2}\int_{-Mt}^{Mt}(% \varphi_{N}(\tau)+\mathrm{i}N^{-1+\xi}\varphi^{\prime}_{N}(\tau))\partial_{% \tau}\frac{m_{\mathrm{fc},t}(\tau+\mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{% \prime})}{\tau+\mathrm{i}N^{-1+\xi}-z^{\prime}+t(m_{\mathrm{fc},t}(\tau+% \mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{\prime}))}\\ &\displaystyle\quad\times\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(\tau+% \mathrm{i}N^{-1+\xi})-m_{\mathrm{fc},t}(z^{\prime})}{\tau+\mathrm{i}N^{-1+\xi}% -z^{\prime}}\,\mathrm{d}\tau;\end{split} (6.174)

for (2), we have the sum:

 \begin{split}\displaystyle I_{2,+,2}&\displaystyle=-t^{2}\varphi_{N}(-tM)\int^% {N^{10C_{V}}}_{N^{-1+\xi}}\partial_{\tau}\frac{m_{\mathrm{fc},t}(-tM+\mathrm{i% }\eta)-m_{\mathrm{fc},t}(z^{\prime})}{-tM+\mathrm{i}\eta-z^{\prime}+t(m_{% \mathrm{fc},t}(-tM+\mathrm{i}\eta)-m_{\mathrm{fc},t}(z^{\prime}))}\\ &\displaystyle\quad\times\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(-tM+% \mathrm{i}\eta)-m_{\mathrm{fc},t}(z^{\prime})}{-2tM+\mathrm{i}\eta-z^{\prime}}% \mathrm{d}\eta,\end{split} (6.175)

for (3):

 \begin{split}\displaystyle I_{2,+,3}&\displaystyle=t^{2}\varphi_{N}(tM)\int^{N% ^{10C_{V}}}_{N^{-1+\xi}}\partial_{\tau}\frac{m_{\mathrm{fc},t}(tM+\mathrm{i}% \eta)-m_{\mathrm{fc},t}(z^{\prime})}{tM+\mathrm{i}\eta-z^{\prime}+t(m_{\mathrm% {fc},t}(tM+\mathrm{i}\eta)-m_{\mathrm{fc},t}(z^{\prime}))}\\ &\displaystyle\quad\times\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(tM+% \mathrm{i}\eta)-m_{\mathrm{fc},t}(z^{\prime})}{tM+\mathrm{i}\eta-z^{\prime}}% \mathrm{d}\eta.\end{split} (6.176)

We similarly define I_{2,-,k}, k=1,2,3 as the line integrals along the boundary of \Omega_{N}^{**}\cap\{\mathrm{Im}\mbox{ }z<0\}. We now insert I_{2,\pm,k}(z^{\prime}) into the integral (6.141) and apply Green’s theorem to obtain

 \begin{split}&\displaystyle\frac{1}{2i}\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}% \mbox{ }z^{\prime}>0\}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})I_{2,% \pm,k}(z^{\prime})\,\mathrm{d}z^{\prime}\\ \displaystyle=&\displaystyle\frac{1}{4}\int_{-qG}^{qG}(\varphi_{N}(s)+\mathrm{% i}N^{-1+\xi}\varphi_{N}^{\prime}(s))I_{2,\pm,k}(s+\mathrm{i}N^{-1+\xi})\mathrm% {d}s\\ \displaystyle+&\displaystyle\frac{1}{4}\int^{N^{-1+\xi}}_{N^{10C_{V}}}\varphi_% {N}(-qG)I_{2,\pm,k}(-qG+\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{\prime}\\ \displaystyle+&\displaystyle\frac{1}{4}\int_{N^{-1+\xi}}^{N^{10C_{V}}}\varphi_% {N}(qG)I_{2,\pm,k}(qG+\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{\prime}\\ \displaystyle:=&\displaystyle I_{2,\pm,k,+,1}+I_{2,\pm,k,+,2}+I_{2,\pm,k,+,3}.% \end{split} (6.177)

Applying Green’s theorem to the z^{\prime} integral in the region \Omega_{N}^{*}\cap\{\mathrm{Im}\mbox{ }z^{\prime}<0\}, we obtain:

 \begin{split}&\displaystyle\frac{1}{2i}\int_{\Omega_{N}^{*}\cap\{\mathrm{Im}% \mbox{ }z^{\prime}>0\}}\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})I_{2,% \pm,k}(z^{\prime})\,\mathrm{d}z^{\prime}\\ \displaystyle=&\displaystyle\frac{1}{4}\int_{qG}^{-qG}(\varphi_{N}(s)-\mathrm{% i}N^{-1+\xi}\varphi_{N}^{\prime}(s))I_{2,\pm,k}(s-\mathrm{i}N^{-1+\xi})\mathrm% {d}s\\ \displaystyle+&\displaystyle\frac{1}{4}\int_{-N^{-1+\xi}}^{-N^{10C_{V}}}% \varphi_{N}(-qG)I_{2,\pm,k}(-qG-\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{% \prime}\\ \displaystyle+&\displaystyle\frac{1}{4}\int^{-N^{-1+\xi}}_{-N^{10C_{V}}}% \varphi_{N}(qG)I_{2,\pm,k}(qG-\mathrm{i}\eta^{\prime})\,\mathrm{d}\eta^{\prime% }\\ \displaystyle:=&\displaystyle I_{2,\pm,k,-,1}+I_{2,\pm,k,-,2}+I_{2,\pm,k,-,3}.% \end{split} (6.178)

So far, we have

 \displaystyle\frac{t^{2}}{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{**}}% \partial_{\bar{z}}\tilde{\varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(% z^{\prime})\partial_{z}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})\partial_{z^{% \prime}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-z^{\prime}% }\,\mathrm{d}z\mathrm{d}z^{\prime} \displaystyle= \displaystyle\sum_{k=1}^{3}\sum_{j=1}^{3}\sum_{\alpha,\beta\in\{\pm\}}I_{2,% \alpha,k,\beta,j}.

The main contribution comes from the terms I_{2,\pm,1,\pm,1}. The remaining terms are polynomially smaller.

###### Proposition 6.15.

For any choice of k,j with (k,j)\neq(1,1), \alpha,\beta\in\{\pm\},

 |I_{2,\alpha,k,\beta,j}|=\mathcal{O}(1).

Proof.   We start with k=1, j=2,3. By symmetry, it suffices to deal with j=2. That is, we estimate

 \displaystyle I_{2,\pm,1,\pm,2} \displaystyle= \displaystyle\pm\frac{\varphi_{N}(qG)}{4}\int_{N^{-1+\xi}}^{N^{10C_{V}}}\int_{% -Mt}^{Mt}\frac{\varphi_{N}(\tau)\pm\mathrm{i}N^{-1+\xi}\varphi_{N}^{\prime}(% \tau)}{1-tR_{2}(\tau\pm\mathrm{i}N^{-1+\xi})}S_{2,1}(\tau\pm\mathrm{i}N^{-1+% \xi},qG+\mathrm{i}\eta^{\prime})\,\mathrm{d}\tau\mathrm{d}\eta^{\prime}. (6.179)

Compute the kernel

 \displaystyle\frac{1}{1-tR_{2}(\tau\pm\mathrm{i}N^{-1+\xi})}S_{2,1}(\tau\pm% \mathrm{i}N^{-1+\xi},qG+\mathrm{i}\eta^{\prime}) \displaystyle= \displaystyle-t^{2}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})-m_% {\mathrm{fc},t}^{\prime}(z)(z-z^{\prime})}{(z-z^{\prime}+t(m_{\mathrm{fc},t}(z% )-m_{\mathrm{fc},t}(z^{\prime}))^{2}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc}% ,t}(z^{\prime})-m_{\mathrm{fc},t}^{\prime}(z^{\prime})(z-z^{\prime})}{(z-z^{% \prime})^{2}}

where z=\tau\pm\mathrm{i}N^{-1+\xi}, z^{\prime}=qG+\mathrm{i}\eta^{\prime}. Since |\tau|\leq 2Mt, we have

 |\tau-qG|\geq(1/2)qG\geq t^{1/2}N^{\sigma/2},

and so, as in Proposition 6.13,

 \left|t^{2}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})-m_{\mathrm% {fc},t}^{\prime}(z)(z-z^{\prime})}{(z-z^{\prime}+t(m_{\mathrm{fc},t}(z)-m_{% \mathrm{fc},t}(z^{\prime}))^{2}}\right|\leq Ct^{1/2+\sigma/2}.

Inserting this into (6.179), we find

 \displaystyle|I_{2,\pm,1,\pm,2}| \displaystyle\leq \displaystyle C(1+\|\varphi_{N}^{\prime}\|_{L^{\infty}}N^{-1+\xi})t^{1/2+% \sigma/2}\int_{N^{-1+\xi}}^{N^{10C_{V}}}\int_{-Mt}^{Mt}\frac{1}{|\tau+\mathrm{% i}N^{-1+\xi}-qG-i\eta|^{2}}+\frac{|m_{\mathrm{fc},t}^{\prime}(z^{\prime})|}{|% \tau+\mathrm{i}N^{-1+\xi}-qG-i\eta|}\mathrm{d}\tau\mathrm{d}\eta^{\prime}.

Recalling that |m_{\mathrm{fc},t}^{\prime}(z)^{\prime}|\leq Ct^{-1}, this quantity is bounded by

 C\log N(1+\|\varphi_{N}^{\prime}\|_{L^{\infty}}N^{-1+\xi})t^{1/2+\sigma}N^{% \sigma}.

We have shown

 |I_{2,\pm,1,\pm,2}|,|I_{2,\pm,1,\pm,3}|=\mathcal{O}(\log N(1+\|\varphi_{N}^{% \prime}\|_{L^{\infty}}N^{-1+\xi})t^{1/2+\sigma}N^{\sigma}). (6.180)

We now estimate I_{2,\pm,2,\pm,1}:

 t^{2}\varphi_{N}(-Mt)\int^{N^{10C_{V}}}_{N^{-1+\xi}}\int_{-qG}^{qG}(\varphi_{N% }(s)\pm\mathrm{i}N^{-1+\xi}\varphi_{N}^{\prime}(s))\frac{1}{1-tR_{2}(-Mt\pm% \mathrm{i}\eta)}S_{2,1}(-Mt+\mathrm{i}\eta,s\pm\mathrm{i}N^{-1+\xi})\,\mathrm{% d}s\mathrm{d}\eta. (6.181)

The kernel is

 \begin{split}&\displaystyle\frac{1}{1-tR_{2}(-tM\pm i\eta)}S_{2,1}(-tM+\mathrm% {i}\eta,s\pm\mathrm{i}N^{-1+\xi})\\ \displaystyle=&\displaystyle-t^{2}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}% (z^{\prime})-m_{\mathrm{fc},t}^{\prime}(z)(z-z^{\prime})}{(z-z^{\prime}+t(m_{% \mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))^{2}}\frac{m_{\mathrm{fc},t}(z% )-m_{\mathrm{fc},t}(z^{\prime})-m_{\mathrm{fc},t}^{\prime}(z^{\prime})(z-z^{% \prime})}{(z-z^{\prime})^{2}},\end{split} (6.182)

with z=-qG\pm\mathrm{i}\eta and z^{\prime}=s\pm\mathrm{i}N^{-1+\xi}. The I_{2,\pm,2,\pm,1} can be performed in the same way whether \mathrm{Im}\mbox{ }z\mathrm{Im}\mbox{ }z^{\prime}>0 or \mathrm{Im}\mbox{ }z\mathrm{Im}\mbox{ }z^{\prime}<0, except in the region

 \{(z,z^{\prime}):|s+tM|\leq Mt/2,N^{-1+\xi}<\eta

If |s+tM|\geq Mt/2 and \eta\geq Mt/10, we use the estimate (6.168)

 \frac{1}{|z-z^{\prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))|}% \leq\frac{C}{|tM+s|},

and

 |z-z^{\prime}|=|tM-s+\mathrm{i}N^{-1+\xi}+\mathrm{i}\eta+\tau|\geq\eta (6.184)

to find the bound

 \begin{split}&\displaystyle t^{2}\int_{\{\eta:\eta\geq Mt/10\}}\int_{\{s:|tM+s% |\geq Mt/2\}}\frac{(1+|m_{\mathrm{fc},t}^{\prime}(z)||z-z^{\prime}|)(1+|m_{% \mathrm{fc},t}^{\prime}(z^{\prime})||z-z^{\prime}|)}{|z-z^{\prime}+t(m_{% \mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))|^{2}|z-z^{\prime}|^{2}}\,% \mathrm{d}s\mathrm{d}\eta\\ \displaystyle\leq&\displaystyle Ct^{2}\int_{\{N^{-1+\xi}<\eta\leq Mt/10\}}t^{-% 1}(\eta^{-2}+|m_{\mathrm{fc},t}^{\prime}(z)|\eta^{-1}+|m_{\mathrm{fc},t}^{% \prime}(z^{\prime})|\eta^{-3/2}t^{1/2})\,\mathrm{d}s\\ \displaystyle=&\displaystyle\mathcal{O}(1).\end{split} (6.185)

To pass to the last line, we have used |m_{\mathrm{fc},t}^{\prime}(z)|\leq C\eta^{-1}. For the case |s+tM|\leq Mt/2 and \eta\geq Mt/10, we have the bound

 \begin{split}&\displaystyle t^{2}\int_{\{\eta\geq Mt/10\}}\int_{\{s:|tM+s|\leq Mt% /2\}}\frac{(1+|m_{\mathrm{fc},t}^{\prime}(z)||z-z^{\prime}|)(1+|m_{\mathrm{fc}% ,t}^{\prime}(z^{\prime})||z-z^{\prime}|)}{\eta^{2}|z-z^{\prime}|^{2}}\,\mathrm% {d}s\mathrm{d}\eta\\ \displaystyle\leq&\displaystyle Ct^{2}\int_{\{s:|s+tM|\leq tM/2\}}t^{-3}\,% \mathrm{d}\eta\\ \displaystyle=&\displaystyle\mathcal{O}(1).\end{split} (6.186)

If |s+tM|\geq Mt/2 and \eta\leq Mt/10, use |z-z^{\prime}|\geq|s+tM| to find the estimate

 \begin{split}&\displaystyle t^{2}\int_{\{\eta:N^{-1+\xi}<\eta\leq Mt/10\}}\int% _{\{s:|tM+s|\geq Mt/2\}}\frac{(1+|m_{\mathrm{fc},t}^{\prime}(z)||z-z^{\prime}|% )(1+|m_{\mathrm{fc},t}^{\prime}(z^{\prime})||z-z^{\prime}|)}{|tM+s|^{2}|z-z^{% \prime}|^{2}}\,\mathrm{d}s\mathrm{d}\eta\\ \displaystyle\leq&\displaystyle Ct^{-1}\int_{\{\eta:N^{-1+\xi}<\eta\leq Mt/10% \}}(1+t|m_{\mathrm{fc},t}^{\prime}(z)|+t|m_{\mathrm{fc},t}^{\prime}(z^{\prime}% )|)\,\mathrm{d}\eta\\ \displaystyle=&\displaystyle\mathcal{O}(1).\end{split} (6.187)

At this point, we have obtained estimates for I_{2,\pm,2,\pm,1} in the complement of (6.183)

We now estimate the contribution to I_{2,\alpha,2,\beta,1} from the region (6.183), when \alpha and \beta are of opposite signs. This term is somewhat delicate. It will suffice to deal with I_{2,+,2,-,1}. We split the s integral into the regions \{s:|s+Mt|\leq Mt/2\} and its complement. In the first region \varphi_{N}(s)\equiv\varphi_{N}(-Mt). When \eta<Mt/10 as well, we expand the kernel to second order. For this, \mathrm{Im}\mbox{ }z\mathrm{Im}\mbox{ }z^{\prime}<0, so we have the expansion

 m_{\mathrm{fc},t}(z^{\prime})=m_{\mathrm{fc},t}(\bar{z})+m_{\mathrm{fc},t}^{% \prime}(\bar{z})(z^{\prime}-\bar{z})+\mathcal{O}(\max_{z}|m_{\mathrm{fc},t}^{% \prime\prime}(z)||z-\bar{z}|^{2}) (6.188)

Using (6.188) and the lower bound

 |m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(\bar{z})|=2\pi\rho_{\mathrm{fc},t}(z)% \geq c>0,

the kernel (6.182) is given by:

 \displaystyle- \displaystyle\frac{1}{(z-z^{\prime})^{2}}\Big{(}1-\frac{m_{\mathrm{fc},t}^{% \prime}(\bar{z})}{2\pi\mathrm{i}\rho_{\mathrm{fc},t}(z)}(z-\bar{z})-\frac{m_{% \mathrm{fc},t}^{\prime}(z)}{2\pi\mathrm{i}\rho_{\mathrm{fc},t}(z)}(z-z^{\prime% })-\frac{m_{\mathrm{fc},t}^{\prime}(\bar{z})}{2\pi\mathrm{i}\rho_{\mathrm{fc},% t}(z)}(z^{\prime}-\bar{z}) (6.189) \displaystyle\quad+\frac{(m_{\mathrm{fc},t}^{\prime}(\bar{z}))^{2}}{4\pi^{2}% \rho_{\mathrm{fc},t}(z)}(z^{\prime}-\bar{z})(z-\bar{z})+\frac{m_{\mathrm{fc},t% }^{\prime}(\bar{z})m_{\mathrm{fc},t}^{\prime}(z)}{4\pi^{2}\rho_{\mathrm{fc},t}% (z)}(z^{\prime}-\bar{z})(z-z^{\prime})\Big{)} \displaystyle\times\left(1-\frac{m_{\mathrm{fc},t}^{\prime}(\bar{z})}{\pi% \mathrm{i}\rho_{\mathrm{fc},t}(z)}(z^{\prime}-\bar{z})+\frac{z-z^{\prime}}{\pi% \mathrm{i}t\rho_{\mathrm{fc},t}(z)}\right) (6.190) \displaystyle+ \displaystyle\mathcal{O}\left(\frac{\max_{z}|m_{\mathrm{fc},t}^{\prime\prime}(% z)|(|z-z^{\prime}|^{2}+|\bar{z}-z^{\prime}|^{2})}{|z-z^{\prime}|^{2}}\right), (6.191)

for |tM-s|\leq Mt/2 and \eta\leq Mt/10.

The cancellation that arises from performing the s integral first in (6.189), (6.190) is crucial. For example, the contribution to I_{2,+,2,-,1} from \{|s-Mt|\leq Mt/2\}, \eta<Mt/10 of the term 1/(z-z^{\prime})^{2} is

 \displaystyle\varphi_{N}(-Mt)^{2}\int^{Mt/10}_{N^{-1+\xi}}\int_{s:|s+Mt|\leq Mt% /2}\frac{1}{(-Mt+\mathrm{i}\eta-s+\mathrm{i}N^{-1+\xi})^{2}}\,\mathrm{d}s% \mathrm{d}\eta \displaystyle\leq \displaystyle C\varphi_{N}(-Mt)^{2}\int^{Mt/10}_{N^{-1+\xi}}\frac{tM}{(tM/2)^{% 2}+(\eta+N^{-1+\xi})^{2}}\mathrm{d}\eta. \displaystyle= \displaystyle\mathcal{O}(1).

To estimate the remaining terms, letting z=-tM+\mathrm{i}\eta and z^{\prime}=s-\mathrm{i}N^{-1+\xi}, we compute:

 \displaystyle\int_{s:|s+Mt|\leq Mt/2}\frac{z^{\prime}-\bar{z}}{(z-z^{\prime})^% {2}}\mathrm{d}s \displaystyle=-\int_{s:|s-Mt|\leq Mt/2}\frac{1}{z-z^{\prime}}\mathrm{d}z+\int_% {s:|s-Mt|\leq Mt/2}\frac{z-\bar{z}}{(z-z^{\prime})^{2}}\,\mathrm{d}s, (6.192) \displaystyle\int_{s:|s+Mt|\leq Mt/2}\frac{1}{z-z^{\prime}}\mathrm{d}s \displaystyle=\log\left(\frac{\frac{tM}{2}+\mathrm{i}\eta+\mathrm{i}N^{-1+\xi}% }{\frac{tM}{2}-\mathrm{i}\eta-\mathrm{i}N^{-1+\xi}}\right)=\mathcal{O}(\eta/t)% ,\quad|\eta|\leq Mt/10, (6.193) \displaystyle\int_{s:|s+Mt|\leq Mt/2}\frac{z-\bar{z}}{(z-z^{\prime})^{2}}\,% \mathrm{d}s \displaystyle=\frac{2\mathrm{i}\eta tM}{(tM/2)^{2}+(\eta+N^{-1+\xi})^{2}}. (6.194)

We have used the principal determination of the logarithm in (6.193).

Using (6.193), (6.194), and (6.155),

 \displaystyle\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt/2}\left(1+\frac{% 1}{2\pi\mathrm{i}\rho_{\mathrm{fc},t}(z)}\right)\frac{m_{\mathrm{fc},t}^{% \prime}(\bar{z})(z^{\prime}-\bar{z})}{(z-z^{\prime})^{2}}\mathrm{d}s\mathrm{d}\eta \displaystyle= \displaystyle\int_{N^{-1+\xi}}^{Mt/10}\frac{\mathcal{O}(t^{-1}|\eta|tM)}{(tM/2% )^{2}+(\eta+N^{-1+\xi})^{2}}+\mathcal{O}(t^{-2}|\eta|)\mathrm{d}\eta=\mathcal{% O}(1), \displaystyle\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt/2}\frac{m_{% \mathrm{fc},t}^{\prime}(\bar{z})(z-\bar{z})}{(z-z^{\prime})^{2}}\mathrm{d}s% \mathrm{d}\eta \displaystyle= \displaystyle\int^{Mt/10}_{N^{-1+\xi}}\mathcal{O}(\eta)\frac{tM}{(tM/2)^{2}+(% \eta+N^{-1+\xi})^{2}}\mathrm{d}\eta=\mathcal{O}(1), \displaystyle\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt/2}\frac{1}{t(z-z% ^{\prime})}\mathrm{d}s\mathrm{d}\eta=\mathcal{O}(1).

Moreover, for z=-tM+\mathrm{i}\eta, z^{\prime}=s-\mathrm{i}N^{-1+\xi}, we have

 |\bar{z}-z^{\prime}|,|z-\bar{z}|\leq 2|z-z^{\prime}|, (6.195)

so

 \begin{split}&\displaystyle\left|\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt% /2}\frac{(m_{\mathrm{fc},t}^{\prime}(\bar{z}))^{2}}{4\pi^{2}\rho_{\mathrm{fc},% t}(z)}\frac{(z^{\prime}-\bar{z})(z-\bar{z})}{(z-z^{\prime})^{2}}\left(1+m_{% \mathrm{fc},t}^{\prime}(\bar{z})(z^{\prime}-\bar{z})-\frac{z-z^{\prime}}{t}% \right)\mathrm{d}s\mathrm{d}\eta\right|\\ \displaystyle\leq&\displaystyle\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt% /2}t^{-2}(1+t^{-1}(|z-z^{\prime}|+|\bar{z}-z^{\prime}|)\,\mathrm{d}s\mathrm{d}% \eta=\mathcal{O}(1).\end{split} (6.196)

Similar estimates hold for the other terms containing a quadratic expression in z^{\prime}-z, \bar{z}-z^{\prime} or z-\bar{z} in (6.189), (6.190).

For the error term (6.191), we use (6.195) and the estimate

 \begin{split}\displaystyle|\partial_{z}^{2}m_{\mathrm{fc},t}(z)|&\displaystyle% \leq\left|\frac{1}{N}\sum_{j=1}^{N}\frac{g_{j}(z)^{3}(1+t\partial_{z}m_{% \mathrm{fc},t}(z))}{(1-tR_{2}(z))^{2}}\right|\\ &\displaystyle\leq Ct^{-2}.\end{split} (6.197)

The last step in (6.197) follows from (6.167) and (6.37). (See also [55, Lemma 7.2].) The result is

 \displaystyle\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt/2}\frac{\max_{z}% |m_{\mathrm{fc},t}^{\prime\prime}(z)|(|z-z^{\prime}|^{2}+|\bar{z}-z^{\prime}|^% {2})}{|z-z^{\prime}|^{2}}\mathrm{d}s\mathrm{d}\eta \displaystyle\leq \displaystyle C\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM/2|\leq Mt/2}\mathrm{d}s% \mathrm{d}\eta=\mathcal{O}(1).

At this point all terms in the expansion (6.189), (6.190) are accounted for.

To estimate the contribution from the region (6.183) to I_{2,\alpha,2,\beta,1} when \alpha and \beta have the same sign, we use (see (6.165))

 \left|\partial_{z}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime})}{z-% z^{\prime}+t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))}\right|\leq Ct% ^{-2},

and the estimate

 \left|\partial_{z^{\prime}}\frac{m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{% \prime})}{z-z^{\prime}}\right|\leq Ct^{-2}, (6.198)

which follows from (6.98) and the estimate (6.197). We have:

 \displaystyle|I_{2,+,2,+,1}| \displaystyle\leq Ct^{-2}\varphi(-Mt)\int_{N^{-1+\xi}}^{Mt/10}\int_{s:|s+tM|% \leq Mt/2}|\tilde{\varphi}(s)|\,\mathrm{d}\eta\mathrm{d}\varleq C.

The same bound holds for I_{2,-,2,-,1}.

Replacing z=-Mt+\mathrm{i}\eta by z=Mt+\mathrm{i}\eta, we obtain the bounds |I_{2,\pm,3,\pm,1}|=\mathcal{O}(1). Turning to I_{2,\pm,2,\pm,2}, we have to estimate:

 \pm\varphi(-Mt)\varphi_{N}(-qG)\int_{-N^{C10_{V}}}^{-N^{-1+\xi}}\int_{N^{-1+% \xi}}^{N^{10C_{V}}}\frac{1}{1-tR_{2}(-Mt+\mathrm{i}\eta)}S_{2,1}(-tN^{\sigma}% \pm\mathrm{i}\eta,-qG\pm\mathrm{i}\eta^{\prime})\mathrm{d}\eta\mathrm{d}\eta^{% \prime}.

Note that

 |z-z^{\prime}-t(m_{\mathrm{fc},t}(z)-m_{\mathrm{fc},t}(z^{\prime}))|\geq cqG (6.199)

for z=-Mt+i\eta and z^{\prime}=-qG+i\eta^{\prime}, so by (6.156), (6.9) the integrand is bounded by Ct^{1/2+\sigma/2}|\eta|^{-1}|\eta^{\prime}|^{-1}. Performing the double integration, we obtain a bound of

 Ct^{1/2+\sigma/2}(\log N)^{2}.

This last estimate depended only on the lower bound (6.199), so we have the same estimate for I_{2,\pm,2,\pm,3}, I_{2,\pm,3,\pm,2}, I_{2,\pm,3,\pm,3}. ∎

Denote \tau^{\pm}=\tau\pm\mathrm{i}N^{-1+\xi} and s^{\pm}=s\pm\mathrm{i}N^{-1+\xi}. We have shown

 \displaystyle\int\int_{\Omega_{N}\times\Omega_{N}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{1-tR_% {2}(z)}S_{2,1}(z,z^{\prime})\,\mathrm{d}z^{\prime}\mathrm{d}z (6.200) \displaystyle= \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_% {\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}% (s^{+}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+}% )}{\tau-s}\,\mathrm{d}\tau\mathrm{d}s (6.201) \displaystyle+ \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_% {\mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}% (s^{-}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{-}% )}{\tau-s}\,\mathrm{d}\tau\mathrm{d}s (6.202) \displaystyle- \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_% {\mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}% (s^{-}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{-}% )}{\tau-s+2iN^{-1+\xi}}\,\mathrm{d}\tau\mathrm{d}s (6.203) \displaystyle- \displaystyle\frac{t^{2}}{4}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(% \tau)\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_% {\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}% (s^{+}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{+}% )}{\tau-s-2iN^{-1+\xi}}\,\mathrm{d}\tau\mathrm{d}s (6.204) \displaystyle+ \displaystyle\mathcal{O}(1). (6.205)

The main terms here are (6.203) and (6.204). For the remaining terms we have

###### Proposition 6.16.

We have the estimate: There is a constant C such that

 \displaystyle\left|t^{2}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(\tau% )\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{% \mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(% s^{+}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+})% }{\tau-s}\,\mathrm{d}\tau\mathrm{d}s\right| \displaystyle\leq C, (6.206) \displaystyle\left|t^{2}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(\tau% )\tilde{\varphi}_{N}(s)\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{% \mathrm{fc},t}(s^{-})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(% s^{-}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{-})% }{\tau-s}\,\mathrm{d}\tau\mathrm{d}s\right| \displaystyle\leq C. (6.207)

Proof.   We deal with the first quantity. The second quantity is estimated similarly. The kernel part of the integrand is

 -t^{2}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{fc},t}(s^{+})-\partial_{% \tau}m_{\mathrm{fc},t}(\tau^{+})(\tau-s)}{(\tau-s+t(m_{\mathrm{fc},t}(\tau^{+}% )-m_{\mathrm{fc},t}(s^{+})))^{2}}\frac{m_{\mathrm{fc},t}(\tau^{+})-m_{\mathrm{% fc},t}(s^{+})-\partial_{s}m_{\mathrm{fc},t}(s^{+})(\tau-s)}{(\tau-s)^{2}}.

In the region \{\tau:|\tau-s|\leq Mt\}, we use (6.165), and (6.198). So (6.206) is bounded by

 t^{2}\int_{-Mt}^{Mt}\left(\int_{\tau:|\tau-s|\leq Mt}\frac{C}{t^{4}}\mathrm{d}% \tau+\int_{\tau:|\tau-s|\geq Mt}\frac{(1+|m_{\mathrm{fc},t}^{\prime}(\tau)||% \tau-s|)(1+|m_{\mathrm{fc},t}^{\prime}(s)||\tau-s|)}{|\tau-s|^{4}}\,\mathrm{d}% \tau\right)\mathrm{d}s\leq C. (6.208)

The sum of the remaining terms (6.203), (6.204) is

 -\frac{t^{2}}{2}\int_{-qG}^{qG}\int_{-Mt}^{Mt}\tilde{\varphi}_{N}(\tau)\tilde{% \varphi}_{N}(s)\mathrm{Re}\,\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-})-m% _{\mathrm{fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t% }(s^{+}))}\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{+% })}{\tau-s-2iN^{-1+\xi}}\,\mathrm{d}\tau\mathrm{d}s.

Using the expansion (6.189), (6.190) in the region \{\tau:|\tau|\leq 2Mt\}:

 \displaystyle t^{2}\partial_{\tau}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm% {fc},t}(s^{+})}{\tau-s+t(m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{+}))% }\partial_{s}\frac{m_{\mathrm{fc},t}(\tau^{-})-m_{\mathrm{fc},t}(s^{+})}{\tau-% s-2\mathrm{i}N^{-1+\xi}} \displaystyle= \displaystyle-\frac{1}{(\tau-s-2\mathrm{i}N^{-1+\xi})^{2}}+\Delta_{3},

where \Delta_{3}(z,z^{\prime}) is an error term. The most serious terms in \Delta_{3} are handled using the computation

 \begin{split}&\displaystyle\int_{-2tM}^{2tM}\int_{-tM}^{tM}\frac{m_{\mathrm{fc% },t}^{\prime}(z)}{\rho_{\mathrm{fc},t}(z)}\tilde{\varphi}_{N}(s)\tilde{\varphi% }_{N}(\tau)\frac{1}{s-\tau-2\mathrm{i}N^{-1+\xi}}\,\mathrm{d}s\mathrm{d}\tau\\ \displaystyle=&\displaystyle c\int\int f(\xi)\overline{g(\lambda)}K(\xi-% \lambda)\,\mathrm{d}\xi\mathrm{d}\lambda,\end{split} (6.209)

where f and g are the inverse Fourier transforms of \mathbf{1}_{[-2tM,2tM]}(\tau)\frac{m_{\mathrm{fc},t}^{\prime}(z)}{\rho_{% \mathrm{fc},t}(z}\tilde{\varphi}(\tau) and \mathbf{1}_{[-tM,tM]}(s)\tilde{\varphi}_{N}(s), respectively, and

 \displaystyle K(\xi,\lambda) \displaystyle=K(\xi-\lambda) \displaystyle:=\mathrm{i}\mathbf{1}_{(-\infty,0]}(\xi-\lambda)\mathrm{e}^{-2N^% {-1+\xi}|\xi-\lambda|},

so that

 \widehat{K}(x)=\frac{1}{x-2\mathrm{i}N^{-1+\xi}}.

From the Fourier representation, the Plancherel theorem and the simple estimates

 \|f\|_{L^{2}}=\mathcal{O}(t^{-1/2}),\|g\|_{L^{2}}=\mathcal{O}(t^{1/2}),

the term (6.209) is O(1). All other error terms are then easily estimated, using z-\bar{z}=\mathrm{i}2N^{-1+\xi} and the trivial bound

 \int_{-2tM}^{2tM}\int_{-tM}^{tM}|\tilde{\varphi}_{N}(s)||\tilde{\varphi}_{N}(% \tau)|\frac{1}{|s-\tau-2\mathrm{i}N^{-1+\xi}|^{2}}\,\mathrm{d}s\mathrm{d}\tau=% \mathcal{O}(\log N).

As in (6.208) contribution from the region \{|\tau|\geq 2Mt\}\subset\{\tau:|\tau-s|\geq Mt\}, as well as the error terms, are \mathcal{O}(1). Adding the contributions from the two main terms, we find:

 \displaystyle\int\int_{\Omega_{N}\times\Omega_{N}}\partial_{\bar{z}}\tilde{% \varphi}_{N}(z)\partial_{\bar{z}}\tilde{\varphi}_{N}(z^{\prime})\frac{1}{1-tR_% {2}(z)}S_{2,2}(z,z^{\prime})\,\mathrm{d}z^{\prime}\mathrm{d}z \displaystyle= \displaystyle\frac{1}{2}\int_{-tM}^{tM}\int_{-2tM}^{2tM}\varphi_{N}(\tau)% \varphi_{N}(s)\frac{(\tau-s)^{2}-N^{-2+2\xi}}{((\tau-s)^{2}+N^{-2+2\xi})^{2}}% \mathrm{d}\tau\mathrm{d}s+\mathcal{O}(1) \displaystyle= \displaystyle\frac{1}{2}\int_{-tM}^{tM}\int_{-2tM}^{2tM}\varphi_{N}(\tau)% \varphi_{N}(s)\mathrm{Re}\partial_{s}\frac{1}{\tau-s+\mathrm{i}N^{-1+\xi}}\,% \mathrm{d}\tau\mathrm{d}s+\mathcal{O}(1) \displaystyle= \displaystyle-\frac{1}{2}\int_{-tM}^{tM}\int\varphi_{N}(\tau)\varphi_{N}^{% \prime}(s)\mathrm{Re}\frac{1}{\tau-s+\mathrm{i}N^{-1+\xi}}\,\mathrm{d}\tau% \mathrm{d}s+\mathcal{O}(1). (6.210)

This is the same quantity as in (6.172), and so this ends the computation of the term (6.141).

It remains to estimate I_{1,3}. Integrating by parts in z and z^{\prime}, we have

 I_{1,3}=\frac{t}{N}\sum_{j=1}^{N}\int_{\Omega_{N}^{*}}\int_{\Omega_{N}^{*}}g_{% j}(z)g_{j}(z^{\prime})\partial_{\tau}\tilde{\varphi}(z)\partial_{s}\tilde{% \varphi}(z^{\prime})\,\mathrm{d}z\mathrm{d}z^{\prime}+\mathcal{O}(N^{-2}). (6.211)

As for I_{1,1} and I_{1,2}, we use Green’s theorem to the domains \Omega_{N}\cap\{\mathrm{Im}\mbox{ }z>0\}, \Omega_{N}\cap\{\mathrm{Im}\mbox{ }z<0\}, \Omega_{N}\cap\{\mathrm{Im}\mbox{ }z^{\prime}>0\}, \Omega_{N}\cap\{\mathrm{Im}\mbox{ }z^{\prime}<0\}. By the support properties of \partial_{s}\tilde{\varphi}(\tau), we only find contributions from the segments [-qG\pm\mathrm{i}N^{-1+\xi},qG\pm\mathrm{i}N^{-1+\xi}]. Denoting \tau^{\pm}=\tau\pm\mathrm{i}N^{-1+\xi}, s^{\pm}=s\pm\mathrm{i}N^{-1+\xi}, the result is

 \begin{split}\displaystyle I_{1,3}&\displaystyle=\frac{t}{4N}\sum_{j=1}^{N}% \int_{-qG}^{qG}\int_{-qG}^{qG}g_{j}(\tau^{+})g_{j}(s^{+})\partial_{s}\tilde{% \varphi}_{N}(\tau^{+})\partial_{\tau}\tilde{\varphi}_{N}(s^{+})\,\mathrm{d}s% \mathrm{d}\tau\\ &\displaystyle+\frac{t}{4N}\sum_{j=1}^{N}\int_{-qG}^{qG}\int_{-qG}^{qG}g_{j}(% \tau^{-})g_{j}(s^{-})\partial_{s}\tilde{\varphi}_{N}(\tau^{-})\partial_{\tau}% \tilde{\varphi}_{N}(s^{-})\,\mathrm{d}s\mathrm{d}\tau\\ &\displaystyle-\frac{t}{4N}\sum_{j=1}^{N}\int_{-qG}^{qG}\int_{-qG}^{qG}g_{j}(% \tau^{+})g_{j}(s^{-})\partial_{s}\tilde{\varphi}_{N}(\tau^{+})\partial_{\tau}% \tilde{\varphi}_{N}(s^{-})\,\mathrm{d}s\mathrm{d}\tau\\ &\displaystyle-\frac{t}{4N}\sum_{j=1}^{N}\int_{-qG}^{qG}\int_{-qG}^{qG}g_{j}(% \tau^{-})g_{j}(s^{+})\partial_{s}\tilde{\varphi}_{N}(\tau^{-})\partial_{\tau}% \tilde{\varphi}_{N}(s^{+})\,\mathrm{d}s\mathrm{d}\tau+\mathcal{O}(N^{-2}).\end% {split} (6.212)

By (6.167), we have

 \left|\frac{t}{N}\sum_{j=1}^{N}g_{j}(z)g_{j}(z^{\prime})\right|\leq C,

so from (6.212), we obtain

 |I_{1,3}|\leq C\|\varphi_{N}^{\prime}\|_{L^{1}}^{2}\leq C. (6.213)

Combining the results (6.172), (6.210), (6.213) we find

 \displaystyle V(\varphi_{N}) \displaystyle=\frac{2}{\pi^{2}}(-I_{1,1}+I_{1,2}+I_{1,3})