Spectra of Random Hermitian Matrices with a Small-Rank External Source: The critical and near-critical regimes
Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the external source matrix has two distinct real eigenvalues: with multiplicity and zero with multiplicity . For a Gaussian potential, it was shown by Péché  that when is fixed or grows sufficiently slowly with (a small-rank source), eigenvalues are expected to exit the main bulk for large enough. Furthermore, at the critical value of when the outliers are at the edge of a band, the eigenvalues at the edge are described by the -Airy kernel. We establish the universality of the -Airy kernel for a general class of analytic potentials for for .
- 1 Introduction
- 2 The perturbed equilibrium measure and the local coordinate
- 3 Initial analysis of the Riemann-Hilbert problem: the global parametrix
- 4 The local parametrix near
- 5 The error analysis
- 6 The kernel near the critical region
- A The asymptotic expansion of
Fix a Hermitian matrix . Equip the space of Hermitian matrices with the probability measure
where is the entry-wise Lebesgue measure and the integration is over all Hermitian matrices. The eigenvalues of represent the energy levels of a system without time-reversal invariance . When the external field is nonzero and , this measure arises in the study of Hamiltonians that can be written as the sum of a random matrix and a deterministic source matrix .
When (no external source) and , (1-1) describes the Gaussian Unitary Ensemble, or GUE. For reasonable choices of the spectrum tends to accumulate on fixed bands on the real axis. Introducing an external field can have the effect of perturbing the expected position of the spectrum. For example, Aptekarev, Bleher, and Kuijlaars [16, 3, 17] studied the Gaussian case when the matrix has two eigenvalues , each of multiplicity . When is sufficiently small (the subcritical case), the eigenvalues of accumulate with probability one on a single interval, just as when . As increases the interval splits into two (the supercritical case). There is a transitional value of between these two cases (the critical case) where the local eigenvalue density near where the bands are about to split is described by the Pearcey process. See  and  for further studies of large-rank external field models, and  for recent results on the universality of the Pearcey process.
We are interested instead in small-rank sources of the form
assuming that , . The ratio of to , which is asymptotically small, will be denoted as
The limiting distribution of the largest eigenvalue for such a small-rank external source in the Gaussian case, , was studied by Péché . Again three distinct behaviors were observed. For sufficiently close to zero, i.e. the subcritical case, the largest eigenvalue is expected to lie at the right band endpoint and behave as the largest eigenvalue of an GUE matrix. For large enough, i.e. the supercritical case, eigenvalues are expected to exit the bulk and be distributed as the eigenvalues of an GUE matrix. In the transitional critical case, when the outliers lie very near the band endpoint, the distribution for the largest eigenvalue is an extension of the standard GUE Tracy-Widom function  when (see also [4, 5, 1]). These functions, denoted by , were first discovered by Baik, Ben-Arous, and Péché  in the context of sample covariance, or Wishart, matrices and were shown to be probability distributions by Baik . Adler, Delépine, and van Moerbeke  showed that these distributions also appear when non-intersecting Brownian motions start from when with conditioned to end at when and conditioned to end at when . The walkers all start out in a single group, but at a critical time depending on , a group of walkers separates from the main bulk. At this critical time the walkers on the edge where separation is about to occur follow the -Airy process, which is connected with . See  for other processes in which the functions arise.
In this paper we extend Péché’s result in the critical case to more general functions . Our specific assumptions are listed in Section 1.3, but we essentially assume is a generic analytic potential with sufficient growth at infinity. This establishes a new universality class of matrix ensembles with the local eigenvalue density near the critical point described by the -Airy process. The universality of the supercritical and subcritical cases has been considered separately .
In the case of rank one perturbation (i.e. ), the recent paper  by Baik and Wang has described the limiting distribution of the largest eigenvalue for all the possible cases including the critical case that we consider here.
1.1 The kernel and its connection to multiple orthogonal polynomials
Let be the probability density that the matrix chosen using (1-1) has eigenvalues (here ). Then, when the are distinct, the -point correlation function is . Brézin and Hikami [18, 19, 20, 21] showed that in the Gaussian case, the -point correlation functions can all be expressed in terms of a single kernel :
Zinn-Justin [36, 37] extended this result to the case of more general . We will find the leading term in the large- asymptotic expansion of the kernel in the critical regime near the critical endpoint.
Bleher and Kuijlaars  showed that the kernel can be written in terms of multiple orthogonal polynomials. Furthermore, these multiple orthogonal polynomials can be written in terms of the solution to a certain Riemann-Hilbert problem. Specifically, suppose is a matrix-valued function of the complex variable satisfying
Here denote the non-tangential limits of as approaches the real axis from the upper and lower half-planes. Whenever posing a Riemann-Hilbert problem we assume (unless otherwise stated) that the solution has uniformly Hölder continuous boundary values with any exponent along the jump contour when approached from either side. Under our assumption (iv) in Section 1.3, the unique solution can be written explicitly in terms of multiple orthogonal polynomials of the second kind (see , Section 2). In the case of two distinct eigenvalues and , which is our case, the kernel may be written in terms of the function as
To analyze the asymptotic behavior of we will use the standard nonlinear steepest descent method for Riemann-Hilbert problems, as well as certain ideas introduced by Bertola and Lee [10, 11] to study the first finitely many eigenvalues in the birth of a new spectral band for the random Hermitian matrix model without source.
A potential alternate method for establishing universality for finite would be to use Baik’s result  writing the kernel in terms of the standard (not multiple) orthogonal polynomials. The rank of the matrices in this alternate expression grows with , whereas the size of the Riemann-Hilbert problem (1-5) grows with the number of distinct eigenvalues. As such, for growing it is more convenient to analyze the Riemann-Hilbert problem for multiple orthogonal polynomials.
1.2 Definition of the critical regime
where is the unique measure minimizing the functional
Here is a small parameter which is identified with the ratio . The variational equations are equivalent to the statement that  there exists a real constant such that
We shall denote
Our first assumption will be
It has been shown in  at Theorem 1.3555In the theorem, the changing parameter is essentially after rescaling. that, for real-analytic ,
If is regular for then is still regular for small enough, and
The locations of the spectral edges (the ’s and ’s) are real-analytic functions of such that the bands of the support of the equilibrium measure stay separated as ranges in a small open set around .
In addition it is also known that the support (under the real–analyticity assumption) consists of a finite union of bounded intervals; we will denote the support of the density by (see Figure 2)
We will consider the unperturbed density to be the solution of the above variational problem with ; in this case the reference to will be tacitly suppressed, and so , etc. Define for the unperturbed () problem the following quantities:
Note that and hence ; we choose such that .
It is also known that is a continuous function on and harmonic (and convex) on the complement of the support (up to a sign it is also known as the logarithmic potential in potential theory).
Define to be the (unique) value of so that (here ).
The uniqueness is promptly seen because ; in fact the effective potential is known  to satisfy
In particular, and hence . Thus the critical value of is given by
We also recall that for regular potentials the behavior of (a suitable branch of) the function near any of the endpoints of the interval of support is
for some constant . For the point one can also prove that ; this allows us to introduce the scaling coordinate near via the definition
We now define the critical and near-crtical regimes. A more extensive context for these definition can be found in . For completeness we also define the supercritical, subcritical, and jumping outlier regimes. The supercritical and subcritical regimes are dealt with separately in ; we plan to consider the (non-generic) jumping outlier regime in a future work.
We define, for , to be the unique point on the real axis greater than such that . For we can choose .
The model is in the supercritical regime if has a unique global maximum on at a point and any of the three conditions below is satisfied:
and for some .
and for some .
Note that is always greater than and . If the global maximum on is attained at several distinct points then we will say that we are in the jumping outlier regime.
The matrix model specified by (1-1) is in the subcritical regime if and for all .
1.3 Assumptions and results
We will make the following assumptions on , , and :
Regarding assumption (i), the case when is equivalent by sending and . As for assumption (ii), in the general case when has distinct eigenvalues the kernel can be written in terms of multiple orthogonal polynomials associated to an Riemann-Hilbert problem, which is beyond the scope of this paper. The assumption of analyticity in (iii) allows us to use the nonlinear steepest-descent method for Riemann-Hilbert problems. The assumption of regularity ensures that the equilibrium measure of has square-root decay at each band endpoint (so that we can use Airy parametrices) and that these endpoints are analytic functions of near .
Next, (iv) guarantees the existence of the multiple orthogonal polynomials needed to ensure the Riemann-Hilbert problem has a solution. We note that the allowed include any convex (see the introduction of ).
We compute the large- behavior of the kernel function (1-6) in the critical regime. We explicitly compute the kernel in a neighborhood of . In the remaining portions of the complex plane, our result is that the kernel function converges to the kernel for the classical orthogonal polynomial problem with respect to . That is, away from the standard universality classes apply (i.e. the sine kernel in the bulk of the spectrum and Airy kernels at the other edges). Our main result is:
By dropping the drift term in (1-20) we would simply deteriorate the error estimate to which is however still vanishing since .
The constant admits an explicit integral representation for an arbitrary real-analytic potential but it is a bit complicated when the equilibrium measure is supported on multiple intervals. In the simplest case where the support of the equilibrium measure consists of a single interval then we have
where the contour of integration is a simple closed contour surrounding the support in the complex plane. We will not be using in any way the explicit form of , except the fact that it is a well–defined quantity due to the smoothness of guaranteed by the already cited Kuijlaars’ Theorem 1.3 in .
Acknowledgments. The authors would like to thank Jinho Baik, Ken McLaughlin, and Dong Wang for several illuminating discussions. M. Bertola was supported by NSERC. R. Buckingham was supported by the Charles Phelps Taft Research Foundation. V. Pierce was supported by NSF grant DMS-0806219.
2 The perturbed equilibrium measure and the local coordinate
To describe a growing number of outliers, we define a perturbed equilibrium measure problem with parameter of perturbation for . Using the -function (1-7) we define
The other constant will be defined by (2-7) after we define the locally holomorphic function .
We also note that we have
uniformly over compact subsets of . The ’s () are defined by the same equations as the ’s except that is set to zero (hence all the log terms are dropped) and that is replaced by . Such convergence is uniform (outside a finite disk around and inside a compact set) according to .
Since is regular for , it will still remain regular for small values of . Hence there exists a holomorphic function in a finite disk around such that
for some constant .
Previously, we have defined at . To describe the effect of growing we introduce a more exact definition . Note that
is a locally holomorphic function at because is real analytic.
Definition 2.1 ( and ).
For , is such that at , i.e.
We also define such that
where is Euler’s Gamma function and
One finds explicitly
Since for we have , for sufficiently small we still have .
From Definition 2.1, we have at (because ). For other values of , since only the term in depends on , we get .
For the subsequent exposition, we redefine in a way that is compatible with the earlier Definition 1.2.
Definition 2.2 (replacing Definition 1.2).
From (2-4) and the above definition, we get
3 Initial analysis of the Riemann-Hilbert problem: the global parametrix
We define the contour as the positively-oriented circle centered at (the leftmost edge of the spectrum) and passing through (the rightmost edge of the spectrum). We choose the circle large enough so that is negative on the real axis to the left of . Until further notice the dependence of the various quantities on (i.e. , , , etc.) will be understood throughout. We have:
For sufficiently small and , the function increases as one follows in either direction starting from (i.e. through the upper half-plane or the lower half-plane).
From the definition (2-1), we have . It is obvious that increases when . It is also simple to see that increases along the referred contour because increases along the contour for any in . ∎
We now open up lenses around each of the bands in the standard way (as in the analysis of the orthogonal polynomials associated to ). We introduce the following open regions (see Figure 2):
, : The area in the upper half-plane () or lower half-plane () between the band and its appropriate adjacent lens.
: The area in the upper half-plane () or lower half-plane () between the band and the appropriate adjacent lens.
: The area in the upper half-plane () or lower half-plane () inside the contour but outside the lenses.
: The part of the upper half-plane () or lower half-plane () outside the contour .
We also define
With these definitions of regions and contours, we define in each region by
Then satisfies the following jump conditions (see Figure 2 for the orientation of the contours):
and the boundary condition
Here we have defined by which is some constant on each gap.
We will show below that the above Riemann-Hilbert problem is exponentially close to a simpler one away from the turning points such as . To be more precise, let us define a shrinking disk centered at by
Here we have chosen the diameter of the disk so that (refer to (2-12))
To complete the error analysis we will also need a fixed-size disk around . Fix a small and define to be the finite disk centered at with fixed radius . For large enough, we have . We will show in Section 5 that, for outside and a finite distance away from the other turning points (the ’s, ’s, and ), the Riemann-Hilbert problem for is exponentially close to that for the outer-parametrix as with the proper choices of and . The necessary data on the effective potentials , , and is contained in Lemma 5.5. Thus we propose the following model Riemann-Hilbert problem for the outer parametrix.
In the critical case, the outer parametrix is defined as the solution of the Riemann-Hilbert problem
The function is defined below the equation (3-6).
This Riemann-Hilbert problem is essentially . The unique solution is given in  Lemma 4.3, or in  Lemma 4.6 in a slightly more generalized setup. We refer the interested reader to those papers since this is not essential here. In the subsequent analysis we will only need the following information.
Then for , there is a unique holomorphic matrix-valued function with determinant one such that
as . In addition, has a limit as and
A direct check shows that has the same jumps as in a neighborhood of . Thus the ratio has no jump discontinuities inside and hence may have at most an isolated singularity at . Furthermore, this product has at worst a square-root singularity at (coming from the product of the quarter-root singularities in and ). In the absence of a branch cut, this means is holomorphic. Finally, (3-13) follows from the definition of in (2-3) and the dependence of on . ∎
4 The local parametrix near
We begin this section by expressing the Riemann-Hilbert problem satisfied by inside in terms of the local coordinate . Zooming in on , the contours are shown in Figure 4. There we collapse the global contours and a part of into and . The regions II and III are parts of the region .
Using the identities below,
we see that satisfies the following Riemann-Hilbert problem inside the disk :
We construct below a local parametrix that solves a Riemann-Hilbert problem similar to the above.
4.1 -Airy parametrix
The th derivative of the standard Airy function admits the contour integral representation
where the contour is shown in Figure 4. Extending the standard Airy function, we will need the following generalized Airy functions.
Let us define the following generalized Airy functions corresponding to each contour in Figure 4.