CauchyLaguerre twomatrix model and the MeijerG random point field
M. Bertola, M. Gekhtman, J. Szmigielski
CRM, Université de Montréal and Department of Mathematics and Statistics, Concordia University
Department of Mathematics, University of Notre Dame
Department of Mathematics and Statistics, University of Saskatchewan
Abstract
We apply the general theory of Cauchy biorthogonal polynomials developed in [6] and [7] to the case associated with Laguerre measures. In particular, we obtain explicit formulæ in terms of MeijerG functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy twomatrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two–level random point field, the MeijerG random field. We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the MeijerG random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble.
Contents
 1 Introduction
 2 The kernels for finite and infinite : MeijerG field
 3 Applications
 4 From Jacobi to CauchyLaguerre biorthogonal polynomials; preliminaries
 5 CauchyLaguerre Biorthogonal polynomials and their kernels
 A The MeijerG functions
 B Correlation functions
 C From the MeijerG to the Bessel field
1 Introduction
The Cauchy twomatrix model, introduced in [6], is a random matrix model defined in terms of a probability measure on the space of pairs of positive semidefinite Hermitean matrices. This probability measure depends on the choice of two scalar functions , called the potentials, and is defined as
(11) 
where stands for the ordinary Lebesgue measure on the real vector space of Hermitean matrices. The potentials are supposed to grow so that . The parameter is a scaling parameter which in the asymptotic regime tends to infinity in such a way that . We will assume henceforth and that .
There are several Hermitean multi matrix models; the most studied, and possibly the first, was introduced in [14]; the interaction, instead of , is which we will refer to as the “ItzyksonZuber” (IZ) interaction. Both models have applications to the counting of colored ribbon graphs on Riemann surfaces. The IZ models are expected to display new universality behaviours in appropriate scaling regimes. Partial results supporting that expectation are appearing (i.e. [13], where the authors compute the scaling behavior of the kernels near special points of transition). We briefly remark that one natural way of generating the interaction is to consider the measure where and is the standard Lebesgue measure on the set of complex matrices , and to integrate out the Gaussian part .
It was shown in [6], using methods of [14], that all correlation functions of the eigenvalues of the two matrices can be computed exactly in terms of certain Cauchy biorthogonal polynomials (BOPs). The latter consist of two sequences of polynomials of exact degree with the defining properties
(12) 
In [7] the algebraic properties of these polynomials were investigated but no concrete example which could be considered “classical” was provided. On the other hand, even before the Cauchy BOPs were introduced, an instance reducible to such polynomials and associated with a classical weight, appeared implicitly in [10] in the study of a (different) biorthogonal Laguerre ensemble, one of several examples of biorthogonal ensembles considered there that allow an explicit computation of correlation functions. In this paper, we apply the formalism developed in [6] and [7] to the model defined by the probability measure (the factor of has been absorbed by an obvious rescaling)
(13) 
with associated biorthogonal polynomials defined by
(14) 
The present paper has three main goals:

obtain explicit formulæ for and related functions;

find explicit formulæ for the correlation functions at finite ;

formulate a scaling limit of the correlation functions near the origin and thus define a limiting random point field; because of their expressions in terms of MeijerG functions, we call this the MeijerG random point field.
In one application of the formalism developed in this paper we express the joint statistics of the fluctuations of the smallest eigenvalues of the two matrices in terms of a Fredholm determinant (Sec. 3). This is followed by a numerical evaluation and plots of the distributions of the smallest eigenvalues. We also perform a simple probe into how the CauchyLaguerre twomatrix model relates to the Laguerre ensemble. To this end we formulate a suitable scaling limit in which we recover the Bessel field, thus showing that in an appropriate regime the spectrum of one of the matrices behaves like the spectrum of the Laguerre ensemble.
Remark 1.1.
We point out that this is the first instance of a coupled matrix model for which one can address directly and rigorously the coupled statistics of eigenvalues in a scaling regime: the IZ multimatrix model is to date far from this level of detail, hampered by the lack of an effective description of all four kernels. As a result only the spectrum of one of the two matrices can be effectively analyzed [13] .
For the sake of comparison we briefly review the pertinent results for onematrix models [19] using, as a prototype, the Gaussian Unitary Ensemble (GUE) with probability measure of size . Let denote the largest eigenvalue of : in the limit the probability that is zero. The fluctuations around this maximum in terms of the rescaled eigenvalues are known to be expressible in terms of a determinantal random point field (DRPF, see the review in Section 1.1) with the Airy kernel [22] on a space of Hermitean matrices
(15) 
The famous TracyWidom result [22] connected the probability that to a special solution (HastingsMcLeod) of the second Painlevé equation as follows
This behaviour is now known [12] to be universal, meaning that the Airy kernel arises in a similar scaling limit near the edge of the support of the limiting distribution of eigenvalues, for a general class of potentials instead of just . Moreover, it is known that the Airy DRPF describes a generic behavior near a “soft edge”. The Laguerre ensemble ( positive definite) possesses a “hard edge” at the origin of the spectrum (zero eigenvalue). The statistics of the smallest eigenvalues is determined by the Bessel DRPF near the origin and the gap probability is related to the third Painlevé equation [23]. This behaviour is also “generic” in the sense that it is stable under small perturbations and occurs whenever a hardedge in the onematrix model is present.
The CauchyLaguerre twomatrix model we are considering in this paper is the benchmark for the behavior of a coupled random matrix model with a hardedge and thus plays the same role as the Laguerre ensemble in relation to onematrix models. We shall see that not only can the model be completely elucidated in terms of special functions, but also its scaling behaviour near the hard edge can be expressed in terms of a DRPF as in (17), with kernels described in terms of the generalized hypergeometric functions of type , somewhat reminiscent of kernels and gap probabilities considered in [11].
In [8] we have shown that the spectrum of each of the matrices in the large limit leads to the same Airykernel universality (or other universality classes that already appear in the onematrix model) as long as the limiting eigenvalue distributions of the spectra does not contain the origin in its support: therefore the largest (and smallest) eigenvalue distributions do not differ from the onematrix case.
By contrast, the model we study in this paper falls outside of that universality class: the limiting eigenvalue density was described in ([1], Sec. 6) and near the origin it behaves like . It is therefore natural to expect both new types of kernels as well as new types of gap distributions (see Remark 3.1 and Fig. 6).
The next section reviews the notions of a determinantal random point field, gap probabilities and their computation in terms of Fredholm determinants. Section 2 contains the results of computations involving special functions: the proofs of these results are in Sections 4 and 5. The appendices contain further results of technical nature and some background material used in the main text.
1.1 Short review of Determinantal Random Point Fields
We review the fundamental notion of a random point field (RPF) following [20]. Let be a topological space, called a configuration space; in our case it shall be equipped with the measure induced from the Lebesgue measure on each copy of so that we can define . A configuration is a locally finite collection of points of . A random point field on is a probability measure on the set of all configurations of points. If is a disjoint union of sets we will call a random field a level random point field. Given a Borel set we denote by the integervalued random variable counting the number of points in . Given disjoint sets and a multiindex one defines the points correlation functions by the formula ( denotes the expectation value)
(16) 
The nontrivial fact is that the collection of correlation functions implicitly defines the probability measure on the space of all possible local configurations [20]. A (twolevel) RPF on is a determinantal RPF (DRPF) if all its correlation functions are determinants (see Definition 3’ in [20]) of the form
(17) 
where the functions are called “kernels” and together they give rise to a single kernel . Thus to define a DRPF it is sufficient to display its kernels: we shall do this for both finite as well as for the scaling limit near the origin.
The eigenvalues of two positive definite matrices (which we denote by and ) constitute an example of such DRPF. Given a determinantal point field and a Borel subset , the associated ”gap probability” is the probability that there are no points in and it is computed as follows. The kernel defines an integral operator on . Then the gap probability is given by (see [20])
(18) 
where is the projection defined by restriction and the determinant is a Fredholm determinant.
2 The kernels for finite and infinite : MeijerG field
We recall the results of [6] (collected and explained in Appendix B, in particular (B.2b)). The correlation functions of the eigenvalues in the Cauchy twomatrix model are expressed as determinants
(21) 
where the kernels are given by (in the notation of (17))
(22a)  
(22b) 
while the kernels are defined in terms of the Cauchy biorthogonal polynomials as:
Our first main result is a compact expression for these four kernels at finite ; to present it we set and define two functions:
(24)  
(25) 
In the above expressions is a contour originating at in the lower halfplane and returning to in the upper halfplane in such a way as to leave all the poles of the functions in the numerator containing the variable inside the contour, while leaving those of the functions of variable outside. Such types of integrals are MellinBarnes integrals and the expressions above are special cases of MeijerG functions (see [2], 5.3, p. 206). Then
Theorem 2.1.
The above theorem is a summary of Theorems 5.5,5.6,5.7 which contain computations of the kernels , whereas the ensuing expressions for the kernels are obtained by a simple rewrite using the definitions (22), (23), (24), (25), and the functions and appearing in Theorems 5.5,5.6,5.7. In particular, and .
2.1 Scaling limit: the MeijerG random point field
In the limit , with the substitutions
Definition 2.1.
Let
(26) 
The contour is a contour of the form in Fig. 7 enclosing all the poles in the numerators of the integrands. Note that are MeijerG functions as in [2] and 16.17 in DLMF ^{2}^{2}2DLMF=”Digital Library of Mathematical Functions”, http://dlmf.nist.gov (see definition in App. A).
Theorem 2.2 (MeijerG twolevel random point field and universality class).
In the scaling limit the correlations of the eigenvalues of are determined by the two–level random point field on the configuration space with the kernels below (in the notation of (17))
where the points in the first copy of will be called the ”” field, and the others the ”” field. The kernels are
(27)  
(28) 
with as in Definition 2.1. The convergence is uniform for within compact sets and the error of the approximation is within .
Proposition 5.2 provides alternative expressions for the kernels in terms of “pointsplit bilinear concomitants”, involving no integration, only derivatives. Section 5.3 is devoted to the proof of Theorem 2.2. We expect the following conjecture to be true.
Conjecture 2.1.
The Meijer random field obtained in the scaling limit in this paper is universal within the class of Cauchy matrix models of the form
with analytic near the origin and the scaling .
3 Applications
The simplest statistical information is the density of eigenvalues both for finite and in the scaling limit, in either case obtained directly from the kernels; for the first matrix (similar expression holds for the second matrix)
(31) 
which follows from the expression in Theorems 2.1, 5.5, 5.6 (see (56), (59)). For large (in fact even for small ’s) and
(32) 
A more effective formula is obtained from Proposition 5.2, which involves only derivatives (the expression is cumbersome, so we have opted here for the integral expression instead). Figure 1 compares the exact density (solid line) with the asymptotic density as per (32).
3.1 The distribution of the smallest eigenvalues
According to the general theory outlined in Section 1.1, the probability that the smallest eigenvalue of is greater than some can be expressed in terms of a Fredholm determinant. Denote by the probability measure (13) on the space of pairs of positivedefinite Hermitean matrices of size . We give here two examples.
Probability that .
Denote by the eigenvalues of and those of . Then our results on the scaling limits of the kernel imply the following
(33) 
where is the integral operator with the kernel defined in (27) restricted to the interval , with a similar definition of (see footnote^{3}^{3}3The superscript in refers to the matrix , with a similar definition for .).
Numerical evaluation using the method in [9] is shown in Figure 3. In Figure 2 we compare the Bessel field with with the MeijerG field with . It is natural to compare these two since they both describe a scaling limit of a random matrix model with a hard edge at the origin. Considering their random matrix origin it is clear that the spectrum of the MeijerG field is more attracted towards the hard edge due to the effect of an attraction exerted by the eigenvalues of the other matrix. See also Section 3.2 and Remark 3.1.
Probability that .
With the same notations as above for the eigenvalues of , our results on the scaling limits of the kernel imply
(34) 
where and is the integral operator on defined in the introduction. Explicitly it reads as follows: denote by a vector of , so that and . Then
(35) 
It is difficult to gain a quantitative understanding of the level of correlation between the two matrices; for this reason we have carried out a numerical computation showing, by the way of example, the quantity , which, in view of their definition, would be identically zero if the spectra were independent (see Fig. 5).
3.2 Convergence to the Bessel field
Let us consider our MeijerG DRPF defined by the kernels in Theorem 2.2.
Theorem 3.1.
In the limit (and fixed) the field becomes the Bessel DRPF (with parameter ) under the rescaling . Under the same rescaling the points of the field (almost surely) do not occupy any bounded set.
To prove the theorem it suffices to show that all the correlation functions involving the field tend to zero uniformly on compact sets, while the correlation functions involving the field alone become the correlation functions of the Bessel DRPF with the standard kernel (428). Specifically
(36) 
where the kernel is defined in Theorem 2.2 (the dependence on of the kernel is not indicated explicitly but can be read off (27)). The correlation functions constructed from the determinants of the kernel in (36) are the same as the correlation functions of the Bessel kernel because the prefactor drops from all the determinants (it amounts to a conjugation of the matrix by a diagonal matrix). The proof of (36) by a direct computation is included in Appendix C, where it is also shown that all the correlation functions involving the field tend to zero uniformly on compact sets of the (rescaled) variables (in particular the kernels and tend to zero, and has a limit, which, however, does not affect the correlation functions).
An immediate consequence of Theorem 3.1 is that the corresponding gap probabilities and also both converge to the gap probability of the Bessel field on .
A similar direct inspection of the expressions in Theorem 2.1 shows that in the scaling limit , ,
(37) 
while tend to zero under the same rescaling^{4}^{4}4The kernel does not tend to zero, but it remains bounded and hence it becomes irrelevant in the correlation functions because it appears in the lower left block.. This means that the eigenvalues of the first matrix near the origin “decouple” from those of the second matrix, and behaves exactly as a onematrix Laguerre ensemble in the scaling limit near the hardedge. This limit (37) of the kernels is equivalent to taking the limit from the CauchyLaguerre model to the MeijerG process (with ) followed by the limit (36). Composition of the two scalings is equivalent, up to a normalization constant, to rescaling the pointfield as .
To explain why the convergence to the Bessel field is intuitively clear, the reader should refer to (13). If scales with , the probability of finding eigenvalues of near the origin is suppressed (see Remark 3.1 and Figure 6): its eigenvalues recede from the origin and do not exert any longer attraction on the eigenvalues of , which now behaves as in a onematrix model with a hard edge and thus falls within the same universality class as the Laguerre ensemble. This intuition is based on the electrostatic interpretation of the probability density as explained in [6].
Remark 3.1.
It is explained in [1, 6] that the limiting (macroscopic) densities of eigenvalues^{5}^{5}5In our definition of the measure (13) the correct macroscopic scaling is to consider the eigenvalues of ; had we defined the measure with then we would consider directly the eigenvalues of the ’s. of on (for ) and on (for ) of the three branches of the algebraic curves below (using the first equation for and fixed, and the second for fixed and ) can be computed from the jumps
(38) 
One can verify that in the first case the behaviour of the densities near the origin is while in the second case one of the densities behaves like (and the other is zero), see Fig. 6.
3.3 Outlook: computation of the gap probabilities and integrable PDEs
Although the formulas (33), (34) do compute the statistics of the lowest eigenvalues, they are transcendental and a connection with a nonlinear ordinary differential equation (or partial DE for (34)) is desirable. It should be pointed out, however, that the numerical computation of Fredholm determinants is not harder (in fact far simpler) than the numerical integration of nonlinear differential equations [9]: the graphs for and in Fig. 3 and 5 are computed in few minutes on a lowend machine using the algorithm explained in [9] and provide more than significant digits (see Fig. 4)^{6}^{6}6A Maple worksheet to compute these determinants is available upon request.. The main approach of Tracy and Widom [22, 23] is to derive Hamiltonian equations for the evaluations of the resolvent at the endpoints of the interval.
A different approach (which has been followed in [4, 5]) relies upon the theory of “integrable kernels” of ItsIzerginKorepinSlavnov (IIKS theory for short) [17] and it relates it to the solution of a Riemann–Hilbert problem. These (matrix valued) kernels are of the general form [16]
(39) 
with the property that they are nonsingular on the diagonal, namely, for all .
It is thus an important step to present the kernels in a form similar to (39). This is the purpose of the expressions in Proposition 5.2. The connection to a Riemann–Hilbert problem, once established, allows one to derive nonlinear PDEs for the gap probabilities: this is the approach of [11] and also [4]. The asymptotic kernels are not of this form (except for the “diagonal” ones): this is not necessarily discouraging, since it was shown in [5] that it is still possible to use to the IIKS theory even for kernels that are not immediately of the form (39).
Preliminary results (in preparation with S. Y. Lee) show that the gap solves a particular case of the third Painlevé transcendent in the variable , much in the same spirit as for the TracyWidom distribution. It is our plan to address the connection with Riemann–Hilbert problems and isomonodromic deformation equations in forthcoming publications.
4 From Jacobi to CauchyLaguerre biorthogonal polynomials; preliminaries
As was observed in [10], the Cauchy biorthogonal polynomials in (14) are related to the classical Jacobi orthogonal polynomials for the weight on . We thank A. Borodin for pointing out this connection whose main point is as follows: consider the bimoment matrix
With the change of variables the integral becomes
(41) 
Notice now that the Hankel moment matrix for the Jacobi polynomials on with weight is given by
(42) 
This immediately implies the following Proposition.
Proposition 4.1.
Let denote the th Jacobi orthogonal polynomial normalized as in (414).^{7}^{7}7Here we define the orthogonality on rather than on . The latter is more customary, but the correspondence between the two versions is a simple affine transformation of the independent variable . Then
(43) 
are the Cauchy biorthogonal polynomials (not orthonormal) associated with densities and and
(44) 
where . This result is valid as long as .
Remark 4.1.
The partition function of the corresponding Cauchy twomatrix model can be easily computed using Proposition 4.1 : in [6] it was shown that if are the norms of the monic polynomials (denoted here provisionally by ), namely, , then . Inspection of the leading coefficients of in Theorem 5.1 shows that
(45) 
Hence
(46) 
Here (and only in the above formula) denotes Barnes’ Gfunction, satisfying the relation .
4.1 The kernels of the correlation functions
The statistics of eigenvalues of is expressible in terms of four kernels that can be expressed in terms of the CBOPs and auxiliary functions. In keeping with the notation of [7] we introduce the auxiliary functions