1. Introduction Multimatrix integrals of various types appear in many mathematical and physical applications, such as combinatorics of graphs, topology, integrable systems, string theory, theory of mesoscopic systems or statistical mechanics on random surfaces.

We will consider here only the case of hermitian matrices for which Q belongs to the U(N)-group.

In many applications “to solve” the corresponding matrix model usually means to reduce the number of variables by explicit integrations over most of the variables in such a way that instead of QN2 original integrations (matrix elements) one would be left in the large N limit only with ∼ N integration variables. In this case the integration over the rest of the variables can be performed, at least in the widely used large N limit, by means of the saddle point approximation. A more sophisticated double scaling limit [1] is also possible (if possible at all) only after such a reduction. The key of success is in the fact that after reduction the effective action at the saddle point is still of the order ∼ N2 whereas the corrections given by the logarithm of determinant of the second variation of the action cannot be bigger than ∼ N (the “entropy” of the remaining variables). The problem is thus reduced to the solution of the “classical” saddle point equations, instead of the “quantum” problem of functional (in the large N limit) integration over the original matrix variables.

This volume represents the most recent trends in the random matrix theory with a special emphasis on the exchange of ideas between physical and mathematical communities. The main topics include:

• • random matrix theory and combinatorics

• • scaling limits; universalities and phase transitions in matrix models

• • topologico-combinatorial aspects of the theory of random matrix models

• • scaling limit of correlations between zeros on complex and symplectic manifolds Most contributions are based on talks and series of lectures given by the authors during the MSRI semester “Random Matrix Models and Their Applications” in Spring 1999, and have an expository or pedagogical style.

One of the basic ideas of the MSRI semester was to bring together the leading experts, both physicists and mathematicians, to discuss the latest results in the theory of matrix models and its applications. The book follows this line: it is divided roughly in half between physics and mathematics. The papers by physicists (G. Cicuta; Ph. Di Francesco; V. Kazakov; G. Mahoux, M. Mehta, J.-M. Normand; P. Zinn-Justin) give an overview of different physical problems in which the random matrix theory plays a decisive role, along with a rich variety of methods and ideas used to solve the problems. This includes enumeration of Feynman graphs on Riemann surfaces in the context of two-dimensional quantum gravity, spin systems on random surfaces, “meander problem” and random foldings, enumeration of knots and links, phase transitions and critical phenomena in random matrix models, interacting matrix models, etc.

Introduction

The purpose of this article is to survey recent interactions between statistical questions and integrable theory. Two types of questions will be tackled here:

• (i) Consider a random ensemble of matrices, with certain symmetry conditions to guarantee the reality of the spectrum and subjected to a given statistics. What is the probability that all its eigenvalues belong to a given subset E? What happens, when the size of the matrices gets very large? The probabilities here are functions of the boundary points Ci of E.

• (ii) What is the statistics of the length of the largest increasing sequence in a random permutation, assuming each permutation is equally probable? Here, one considers generating functions (over the size of the permutations) for the probability distributions, depending on the variable x.

The main emphasis of this article is to show that integrable theory serves as a useful tool for finding equations satisfied by these functions of x, and conversely the probabilities point the way to new integrable systems.

These questions are all related to integrals over spaces of matrices. Such spaces can be classical Lie groups or algebras, symmetric spaces or their tangent spaces. In infinite-dimensional situations, the “ ∞ -fold” integrals get replaced by Fredholm determinants.

During the last decade, astonishing discoveries have been made in a variety of directions. A first striking feature is that these probabilities are all related to Painleve equations or interesting generalizations. In this way, new and unusual distributions have entered the statistical world.

1. Introduction

Suppose that we are selecting n points, p1 , p 2 , … , pn , at random in a rectangle, say R = [0,1] x [0,1] (see Figure 1). We denote by π the configuration of n random points. With probability 1, no two points have same x -coordinates nor -coordinates. An up/right path of π is a collection of points pi1, pi2, … ,pik such that x(pil) < x(pi2) < • • • < x(pik) and y(pi1)< y(pi2) < • • • < y (pik). The length of such a path is defined by the number of the points in the path. Now we denote by ln(π) the length of the longest up/right path of a random points configuration π.

1. Introduction

The remarkable and useful formula has been known for the last two decades; see [Itzykson and Zuber 1980; Mehta 1981; Mehta 1991, Appendix A.5]. Here A and A’ are nxn complex hermitian matrices having eigenvalues x := ﹛x1,…,xn﹜ and respectively, integration is over the n×n complex unitary matrices U with the invariant Haar measure dU normalized such that. The function △(x) is the product of differences of the Xj:

We would like to have a similar formula when A and A’ are n×n real symmetric or quaternion self-dual matrices and the integration is over n×n real orthogonal or quaternion symplectic matrices U, a formula not presently known. These three cases are usually denoted by a parameter (3 taking values 1, 2 and 4 corresponding respectively to the integration over n x n real orthogonal, complex unitary and quaternion symplectic matrices U. We will show that the integral in (1-1) with a measure dU invariant under the appropriate group can be expressed in terms of the eigenvalues and eigenfunctions of a particular hamiltonian. This hamiltonian is closely related to the Calogero model [1969a; 1969b; 1971] where one considers the quantum n-body problem with the hamiltonian.

1. Introduction

This paper is about z-measures, a remarkable two-parameter family of measures on partitions introduced by S. Kerov, G. Olshanski and A. Vershik [Kerov et al. 1993] in the context of harmonic analysis on the infinite symmetric group. In a series of papers, A. Borodin and Olshanski obtained fundamental results on z-measures; see the survey [Borodin and Olshanski 2001] in this volume and also [Borodin and Olshanski 1998]. The culmination of this development is an exact determinantal formula for the correlation functions of the z-measures in terms of the hypergeometric kernel [Borodin and Olshanski 2000]. We mention [Borodin et al. 2000] as one of the applications of this formula. The main result of this paper is a representation-theoretic derivation of the formula of Borodin and Olshanski.

In the early days of z-measures, it was already noticed that they have some mysterious connection to the representation theory of SL(2). For example, a z-measure is in fact positive if its two parameters z and z’ are either complex conjugate z' = z or z,z' ∈ (n, n+1) for some n ∈ ℤ. In these cases z — z' is either imaginary or lies in (—1,1), which is reminiscent of the principal and complementary series of representations of SL(2).

Later, Kerov (private communication) constructed an SL(2)-action on partitions for which the z-measures are certain matrix elements.

1. Introduction Let w be a weight function supported on L (a subset of ℝ) that has finite moments of all orders

With w(x) we associate the Hankel matrix, where i, j = 0 , … , n—1. The purpose of this paper is the determination of

for large n with suitable conditions on w. If L is a single interval, say [—1,1], then the asymptotic form of such determinants was computed by Szego [1918] and later by Hirschmann [1966] for quite general w.

Our main result is as follows. Suppose we replace w(x) by a function given in the form wo(x)U(x) where w0(x) is the weight e-xxv. Then for appropriate functions w, the determinants are given asymptotically as n → ∞ by

and G is the Barnes function (see Section 2).

In Section 2 we establish an identity relating Dn[w], Dn[wo], and a certain Predholm determinant and a description of the Predholm determinant from a “linear statistics” point of view. A computation of Dn[wo] is also included. Then in Section 3 the Coulomb fluid approach is used to compute the asymptotics of the Predholm determinant. This, along with the computation of Dn[wo] allows us to give a heuristic, Coulomb fluid derivation of the formula.