Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

# Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

## Abstract

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogues to the Painlevé III equation, which reduces to the Painlevé III equation itself when the pole is simple.

## 1 Introduction and statement of results

We study unitary invariant random matrix ensembles on the space of positive-definite Hermitian matrices defined by the probability measure

 1Cn(detM)αexp[−nTrVk(M)]dM,α>−1, (1.1)

with

 dM=n∏j=1dReMjj∏1≤i

and

 Cn=∫H+n(detM)αexp[−nTrVk(M)]dM, (1.3)

in the case where the potential has a pole of order , i.e.

 Vk(x)=V(x)+(tx)k. (1.4)

The regular part of the potential is a real analytic function on subject to some constraints which we will detail later. In particular, we will require to be such that, for , the limiting mean density of the eigenvalues as is supported on an interval of the form , with .

It is well known that the eigenvalues of a random matrix drawn from the ensemble (1.1) form a determinantal point process. The joint probability distribution of the eigenvalues is given by

 1Zn,kΔ(x)2n∏j=1xαje−nVk(xj)dxj,Δ(x)=∏1≤i

with the partition function given by

 Zn,k=∫[0,+∞)nΔ(x)2n∏j=1xαje−nVk(xj)dxj. (1.6)

For small, the model (1.1) can be seen as a singular perturbation of the unitary invariant ensemble corresponding to . For , the eigenvalues are pushed away from because of the pole in the potential, and for large , the probability of finding eigenvalues close to is small if is independent of . However, if together with , this repulsion becomes weaker and one expects a transition between a regime where eigenvalues are likely to be found in the vicinity of the origin, and one where eigenvalues are unlikely to be found near the origin.

The effect of singular perturbations of unitary invariant ensembles has been of recent interest [5, 4, 27, 28]. In [5], the singularly perturbed Laguerre Unitary Ensemble (pLUE) was studied, given by the measure

 1Cn(detM)αexp[−nTr(M+tM)]dM, (1.7)

on the space , where is a normalisation constant. A relation between this model and the Painlevé III (henceforth PIII) equation was established for fixed in [5]. In subsequent work [4], a singular perturbation of the Gaussian Unitary Ensemble, which we will refer to as pGUE, was studied, defined by the measure

 1ˆCnexp[−nTr(12M2+t2M2)]dM, (1.8)

on the set of Hermitian matrices .

In [4] the double scaling limit where as of the partition function was analysed using Riemann-Hilbert (RH) techniques. A connection to PIII was also found by relating the orthogonal polynomials associated to the pGUE measure to those of the pLUE measure. In [27], the double scaling limit for the eigenvalue correlation kernel in the pLUE was studied. A limiting kernel was found, defined in terms of a model RH problem associated to a special solution of the PIII equation. This limiting kernel degenerates to the Airy kernel if approaches at a slow rate, and to the Bessel hard edge kernel if at a fast rate. In [28], asymptotics for the partition function in the pLUE, again in terms of a PIII transcendent, were obtained.

In the present work, we will obtain asymptotics for the eigenvalue correlation kernel for a fairly general class of potentials , perturbed with a pole of order , in a double scaling limit where the strength of the perturbation, , goes to zero at an appropriate speed as the size of the matrix, , is taken to infinity. The double scaling limit will be tuned in such a way that the macroscopic behaviour of the eigenvalues in the large limit is the same as in the non-singular case where , but such that the microscopic behaviour of the eigenvalues near is affected by the singularity of the potential. In addition, we will obtain double scaling asymptotics for the partition functions associated to the LUE perturbed with a pole of order , and to the GUE perturbed with a pole of order . Our results will be described in terms of special solutions to a family of systems of ODEs, indexed by , which can be seen as a hierarchy of Painlevé III equations.

### 1.1 Motivations

There are a number of motivations for considering the model (1.1). Firstly, it was observed that critical one-matrix models in which the limiting mean density of eigenvalues vanishes in the bulk of the spectrum or at a higher than generic order at the edge of the spectrum, are in one-to-one correspondence with (super-)Liouville field theories. In this context, it is natural to look for models with higher order critical points in which multiple zeros appear in the bulk or at the edge of the spectral density, as it is known that such higher order critical models correspond to coupling a minimal conformal field theory to (super)-Liouville theory. A review of these facts in the non-supersymmetric case can be found in [12]. Supersymmetric versions followed later in [20] and are nicely reviewed in the appendix of [24]. One-matrix models in which the potential has poles exhibit a new type of critical behaviour. Although at present no conformal field theory analogue of such models is known, it is again natural to study higher order critical points in this context.

A second physical motivation arises in the field of quantum transport and electrical characteristics of chaotic cavities. Here the quantity of interest is the Wigner-Smith time-delay matrix , the eigenvalues of which, , are known as the “partial delay times”. In systems in which the dynamics is chaotic, a RMT approach has been quite successful and one of the central results of this approach is the joint probability density for the inverse delay times, , first obtained in [6, 7]. It takes the form

 P(γ1,…,γn)=1Cn,β|Δ(γ)|βn∏j=1γβn/2je−β2γj, (1.9)

where depends on the symmetries of the system, with a common case. Since many observables may be expressed in terms of , the problem of computing expectation values with respect to the above measure is relevant. In particular the observable

 τW=1nTrQ=1nn∑i=1τi, (1.10)

known as the Wigner time-delay has been considered in the recent work [25]. The partition function for our model (1.1) coincides with the moment generating function for the probability density of in the case and . Although we will not scale with in the present paper, the moment generating function in this model is the partition function for a matrix model which shares the feature of a singular potential with our model (1.1).

Furthermore, in a very recent paper [18], an observable known as the “charge relaxation resistance” defined as

 Rq∝TrQ2(TrQ)2=∑nj=1τ2j(∑nj=1τj)2 (1.11)

was considered. To compute , the approach taken in [18] was to compute the distributions of and separately. The associated moment generating function is the partition function for a perturbed pLUE model with a singularity in the potential of order , thereby demonstrating the physical relevance of the model studied here for .

A third motivating model appears in the field of spin-glasses [2]. Here a model corresponding to the GOE perturbed by a pole of order was analysed for its relation to the distribution of the spin glass susceptibility in the Sherrington-Kirkpatrick (SK) mean-field model. The GUE version of the partition function in such a model relates directly to the partition function in the pLUE, as we will see later on in this paper. One-matrix models in which the potential has poles have also appeared in the context of replica field theories [21].

Finally, our last motivation for this work is to seek a natural candidate for a PIII hierarchy. The notion of a PIII hierarchy has appeared little in the literature; one of the few mentions being [23]. This work was partly motivated by the desire to understand whether the hierarchy proposed in [23] would appear when the order of the pole was increased and if not, what form the alternative hierarchy takes. It appears that the hierarchy of equations which we will obtain is different from the one in [23].

### 1.2 Statement of results

Our main results are the following.

1. We define a hierarchy of higher order PIII equations and prove the existence of special pole-free solutions to it.

2. We obtain double scaling asymptotics for the partition function in the LUE perturbed with a pole of order , in terms of a higher order PIII transcendent. This generalizes the result from [27] for .

3. We obtain double scaling asymptotics for the partition function in the GUE perturbed with a pole of order , . They are again given in terms of higher order PIII transcendents. In the case , such asymptotics were already obtained in [4], but written in a different form.

4. In the double scaling limit, we prove, for and for general , that the eigenvalue correlation kernel near of the model (1.1) tends to a limiting kernel built out of a model RH problem associated to the PIII hierarchy. This extends the result from [27] for and .

We now state our results in more detail.

#### A Painlevé III hierarchy

Given , consider the system of ODEs indexed by ,

 p∑q=0(ℓk−p+q+1ℓk−q−(ℓk−p+qℓk−q)′′+3ℓ′k−p+qℓ′k−q−4uℓk−p+qℓk−q)=τp, (1.12)

for unknown functions and , with

 ℓk+1(s)=0,ℓ0(s)=s2. (1.13)

The ’s are real constants that play the role of times. The equation always results in

 u=−14ℓ2k((ℓ2k)′′−3(ℓ′k)2+τ0). (1.14)

Substituting this expression for in the other equations, we are left with equations for unknowns . We refer to this system of equations as the -th member of the Painlevé III hierarchy. Eliminating for leads to an ODE for of order .

###### Example 1.1

For we have the equation

 ℓ′′1(s)=ℓ′1(s)2ℓ1(s)−ℓ′1(s)s−ℓ1(s)2s−τ0ℓ1(s)+τ1s, (1.15)

which we identify as a special case of the Painlevé III equation, see [13].

###### Example 1.2

If we have a system of two ODEs;

 τ12ℓ1(s)ℓ2(s)−τ0ℓ2(s)2+ℓ′2(s)2ℓ2(s)2−ℓ′1(s)ℓ′2(s)ℓ1(s)ℓ2(s)+ℓ′′1(s)ℓ1(s)−ℓ′′2(s)ℓ2(s)−ℓ2(s)2ℓ1(s)=0, (1.16)

and

 ℓ1(s)2ℓ′2(s)2ℓ2(s)2 −ℓ′1(s)2+sℓ′2(s)2ℓ2(s)−ℓ′2(s)−τ0ℓ1(s)2ℓ2(s)2−sτ0ℓ2(s)−τ2 =2ℓ1(s)2ℓ′′2(s)ℓ2(s)−2ℓ1(s)ℓ′′1(s)+sℓ′′2(s)+2ℓ2(s)ℓ1(s). (1.17)

One can eliminate in order to obtain a single equation of order four for .

We can construct, for any , a special set of solutions to the -th member of the PIII hierarchy in terms of a model RH problem. The function will be of particular importance to us. The model RH problem consists of finding a function satisfying the following properties.

#### RH problem for Ψ

• analytic, with as illustrated in Figure 1. The half-lines can be chosen freely in the upper and lower half plane.

• has continuous boundary values as is approached from the left () or right () side of , and they are related by

 Ψ+(z)=Ψ−(z)(10−eπiα1), z∈Σ1, (1.18) Ψ+(z)=Ψ−(z)(0−110), z∈Σ2, (1.19) Ψ+(z)=Ψ−(z)(10−e−πiα1), z∈Σ3. (1.20)
• As , there exist functions such that has the asymptotic behaviour

 Ψ(z)=(I+1z(q(s)−ir(s)ip(s)−q(s))+O(z−2))z−14σ3Nez1/2σ3, (1.21)

where , with , and where the principal branches of and are taken, analytic off and positive for . The third Pauli matrix is given by .

• As , there exists a matrix , independent of , such that has the asymptotic behaviour

 Ψ(z)=Ψ0(s)(I+O(z))e−(−sz)kσ3zα2σ3Hj, (1.22)

for , where are given by

 H1=I (1.23) H2=(10−eπiα1), (1.24) H3=(10e−πiα1). (1.25)
###### Remark 1.3

In the case , the RH problem for coincides, up to the orientation of the contours, with the model RH problem introduced in [27]. In the case , it is a RH problem which can be solved using Bessel functions [26].

###### Remark 1.4

It is important to note that the function

 Ψ(z,s)H−1jz−α2σ3e(−sz)kσ3 (1.26)

is not analytic at and in fact has a jump across . Indeed, if the asymptotic behaviour in condition (d) of the above RH problem were modified to

 Ψ(z,s)=Ψ0(z,s)e−(−sz)kσ3zα2σ3Hj, (1.27)

with analytic at , then the resulting RH problem would have no solution. A more detailed description of the analytic structure of near the origin can be given as follows. Define

 f2(z,s)=e−z2πi∫−∞0|u|αeue−2(−su)kduu−z, (1.28)

and let be defined by

 Ψ(z,s)=ˆΨ0(z,s)(1f2(z,s)01)e−(−sz)kσ3zα2σ3Hj. (1.29)

Then is an analytic function near , and defined by (1.22) takes the form

 Ψ0(s)=ˆΨ0(0,s)(1f2(0,s)01). (1.30)
###### Remark 1.5

If the model RH problem has a solution, it follows from standard techniques that the solution is unique. Existence of a solution is a much more subtle issue. We will show that the model RH problem is solvable for and .

###### Theorem 1.6

Let , and let be the unique solution of the model RH problem for . Then, the limit

 yα(s)=−2idds[limz→∞zΨ(z,s2)e−z1/2σ3N−1z14σ3]12=−2dds(r(s2)) (1.31)

exists and it is a solution of the equation for in the -th member of the Painlevé III hierarchy, with the parameters given by

 τp=⎧⎪⎨⎪⎩42k+1k2, for p=0,−(−4)k+1αk, for p=k,0, for 0

Moreover, has the following asymptotics as and as ,

 y(s)=−8k2k+1(βk−2−32z0)s2k−12k+1+O(1), s→+∞, (1.33) y(s)=O(s2k−1)+O(s2α+1), s→0,s>0, (1.34)

where we have used the constants

 z0=−(2k−1(k−1)!(2k−1)!!)−22k+1, (1.35) βj=−(−z0)−32−j(2j+1)!!2jj!, (1.36)

with the double factorial.

###### Remark 1.7

In fact we will prove a more general result in the sense that all ’s may be extracted from the model problem for . To this end we write the asymptotic expansion of as as

 Ψ(z)=(I+∞∑j=1Cjz−j)z−14σ3Nez1/2σ3,Cj=(qj(s)−irj(s)ipj(s)−qj(s)). (1.37)

Define the formal power series

 m2(z,s)=∞∑j=1rj(s2)s2j−1zj, (1.38)

and define in addition formal power series in , . The quantities and are defined in terms of and as

 m3(z,s):=−∂sm1−r(s2)(m1+1)s−12∂2sm2 +m2(3r′(s2)−r(s2)22s2−3r(s2)2s2+z), (1.39) m4(z,s):=12(r(s2)sm2−∂sm2), (1.40)

and can be found recursively from the relation

 m1,j=12(m24+m2m3)j−12j−1∑i=0m1,j−im1,i, (1.41)

where denotes the coefficient of in the formal power series of . Note that upon substituting in the expressions for and in terms of and , the right hand side of (1.41) only contains with and therefore gives a well defined recursive relation for .

We now introduce the matrix

 Unknown environment '% (1.42)

where . We then obtain expressions for all in terms of via the relation

 4j∑n=0 ℓj−n(s)(4z)n= (1.43) ((4z)j+1Tr[(∂zK)K−1+s2Kσ−K−1+12zK(sσ+−12σ3)K−1σ−])+, (1.44)

where the notation denotes a projection onto the positive power parts of the power series. The above relation gives for the first few ’s,

 ℓ0(s)=s2, (1.45) ℓ1(s)=−4sr′(s2), (1.46) ℓ2(s)=8s(r(s2)2r′(s2)+2s2r′(s2)2−2r(s2)(r′(s2)+s2r′′(s2))−2r′2(s2)). (1.47)

Note that these expressions are independent of .

###### Remark 1.8

There has appeared a distinct definition of a Painlevé III hierarchy in the literature [23]. The model problem for may be connected with this alternative hierarchy if we choose a different time to act as the independent variable in the ODE. To see this, we generalise the model problem by altering the behaviour at the origin to

 Ψ(z)=Ψ0(s)(I+O(z))e−∑kj=1(−sjz)jσ3zα2σ3Hj, (1.48)

and then define

 ˆΨ(z,s):=Ψ−10(s)Ψ(z−1,s). (1.49)

Using the asymptotic behaviour of in the definition of the Lax pair

 ˆB:=∂s1ˆΨˆΨ−1andˆA:=∂zˆΨˆΨ−1, (1.50)

we find

 ˆB=(zˆv(s1)ˆu(s1)z), (1.51)

and , where are dependent matrices and and are some undetermined functions. The form of these Lax matrices precisely matches those proposed in [23] as forming a Lax pair for a Painlevé III hierarchy. However, in the random matrix model under consideration, it is more natural to work with the variable instead of , and this leads us to a different Painlevé hierarchy.

#### Double scaling limit for the partition functions in the perturbed LUE and GUE

Define the pLUE partition function as

 Missing or unrecognized delimiter for \right (1.52)

and the pGUE partition function as

 ZpGUEn,k,α(t)=1n!∫RnΔ(x)2n∏j=1|xj|2αe−n2(x2j+(tx2)k)dxj,α>−1/2. (1.53)

Whereas for the pLUE and pGUE partition functions with , large asymptotics are described in terms of a special solution to the PIII equation [4, 28], when perturbing a unitary invariant ensemble with a pole of order , the special solutions to the higher order analogues of PIII introduced before will appear.

###### Theorem 1.9

As and in such a way that , we have the asymptotics

 ZpLUEn,k,α(t)=ZpLUEn,k,α(0)exp(12∫sn,t0(r(0)−r(ξ))dξξ)(1+O(n−1)), (1.54) ZpGUEn,k,α(t)=ZpGUEn,k,α(0)exp(12∫sn,t0(−α2−rα−12(ξ)−rα+12(ξ))dξξ) × (1+O(n−1)), (1.55)

where is related to the Painlevé transcendent by

 yα(s)=−2dds(rα(s2)),rα(0)=18(1−4α2).

In other words, we have

 rα(s)=18(1−4α2)−12∫√s0yα(η)dη. (1.56)

#### Double scaling limit of the correlation kernel

In what follows, we consider potentials which are real analytic on and such that

 limx→+∞V(x)log(x2+1)=+∞. (1.57)

The correlation kernel for the eigenvalues in the model (1.1) is expressed using orthogonal polynomials with respect to the weight

 w(x)=xαe−nVk(x)=xαexp[−n(V(x)+(tx)k)] (1.58)

on . Let , be the family of monic polynomials of degree characterised by the relations

 ∫∞0pj(x)pm(x)w(x)dx=hjδjm. (1.59)

The correlation kernel of the determinantal point process (1.5) can be written as

 Kn(x,y)=h−1n−1√w(x)w(y)x−y(pn(x)pn−1(y)−pn(y)pn−1(x)). (1.60)

Important quantities that can be computed directly from are the -point correlation functions. The one-point function, or limiting mean eigenvalue density, is given by

 ρ(x):=limn→∞1nKn(x,x), (1.61)

and describes the macroscopic behaviour of the eigenvalues in the large limit. For fixed , the measure is characterised as the equilibrium measure which minimizes

 Ik(ν)=∬log1|x−y|dν(x)dν(y)+∫Vk(y)dν(y), (1.62)

among all Borel probability measures on . If we let and at the same time , the limiting mean eigenvalue density is not affected by the singular perturbation and is equal to the equilibrium measure obtained from minimising

 I(μ)=∬log1|x−y|dμ(x)dμ(y)+∫V(y)dμ(y), (1.63)

in which the singular part of has been dropped. It is this measure and density we will use in the remainder of the paper and refer to as the equilibrium measure. The measure is characterised by the Euler-Lagrange equations [22], stating that there exists such that

 2∫log|x−y|ψ(y)dy−V(x)=ℓ,x∈suppμ, (1.64) 2∫log|x−y|ψ(y)dy−V(x)≤ℓ,x∈[0,+∞). (1.65)

We will require that is such that the equilibrium measure is supported on a single interval of the form , which implies [8] that its density can be written as

 ψ(x)=12πh(x)√b−xx,x∈[0,b], (1.66)

where is a real analytic function, non-negative on . We will also require that is regular, in the sense that

• the density is positive in the interior of its support , i.e.  for ,

• vanishes like a square root at the endpoint and like an inverse square root at the endpoint , i.e. ,

• the variational inequality (1.65) is strict on .

Define functions in terms of the solution to the model RH problem for as follows. For , let

 (ψ1(z,s)ψ2(z,s)):=Ψ(z,s)(10), (1.67)

and for , we define and in such a way that they are analytic in . This means in particular that

 (ψ1(x,s)ψ2(x,s))=Ψ+(x,s)(1−e−πiα), (1.68)

for .

Then, satisfy the system of equations

 ∂z(ψ1ψ2)=1s⎛⎜⎝ˆa−12ˆbs−12U(s)−is−12ˆbis12(ˆc+ˆas−12U(s)−14ˆbs−1U(s)2)−ˆa+12ˆbs−12U(s)⎞⎟⎠(ψ1ψ2), (1.69)

and they are characterised as the unique solution with the asymptotic behaviour

 (ψ1(z,s)ψ2(z,s))=(I+O(z−1))z−14σ3Nez1/2σ3, (1.70)

as in (and, more generally, for for any ), and

 (ψ1(z,s)ψ2(z,s))=O(1)e−(−sz)kσ3zα2σ3, (1.71)

as for . Here, , , , , with

 a(z,s)=−12∂sb(z,s), (1.72) c(z,s)=(z−u)b(z,s)−12∂2sb(z,s), (1.73)

and

 b(z,s)=4(4z)k+1k∑n=0ℓk−n(s)(4z)n. (1.74)
###### Theorem 1.10

In the double scaling limit where and simultaneously in such a way that , with , we have

 limn→∞1c1n2Kn(−uc1n2,−vc1n2;t)=KPIII(u,v;s), (1.75)

for , where

 KPIII(u,v;s)=eπiαψ1(u;s)ψ2(v;s)−ψ1(v;s)ψ2(u;s)2πi(u−v). (1.76)

The limit is uniform for in compact subsets of .

###### Remark 1.11

For , our limiting kernel is equal to the limiting kernel obtained in [27].

###### Theorem 1.12

The limiting kernel is positive for and , and it has the following limits,

 lims→0KPIII(u,v;s)=Jα(−u,−v), (1.77) lims→+∞s2η3c2KPIII(sη(z0+s−η3uc2),sη(z0+s−η3vc2);s)=A(u,v), (1.78)

where , and and are the Bessel and Airy kernels defined as

 Jα(u,v)=Jα(√u)√vJ′α(√v)−Jα(√v)√uJ′α(√u)2(u−v), (1.79) A(u,v)=Ai(u)Ai′(v)−Ai(v)Ai′(u)u−v