1. Introduction

Consider the length of a longest increasing subsequence in a permutation is an increasing subsequence of length r. If we give SN the uniform probability distribution, becomes a random variable and we want to investigate its distribution. This problem was first addressed by Ulam [1961], who made Monte Carlo simulations and concluded that the expectation E[lN] seems to be of order y/N. The first rigorous result was obtained by Hammersley [1972], who considered the following variant of the problem. Consider a Poisson process in the square [0,1] x [0,1] with intensity α, so that the number M of points in the square is Poisson distributed with mean α. Let and be the x- and y-coordinates of the points in the square. This associates a permutation α ∈ SM with each point configuration, and if we condition M to be fixed, equal to iV say, we get the uniform distribution on SN We see that IM(α) equals the number of points, L(α), in an up/right path from (0,0) to (1,1) through the points, and containing as many points as possible.

1. Introduction

Given a probability measure on a space of matrices, the eigenvalue PDF (probability density function) follows by a change of variables. For example, consider the space o f n x n real symmetric matrices A = [aj,k]0≤j,k < n with probability measure proportional to

The eigenvalues are introduced via the spectral decomposition A = RLRT where R is a real orthogonal matrix with columns given by the eigenvectors of A and L = diag Since the change of variables is immediate for the weight function; however the change of variables in (dA) cannot be carried out with such expedience.

The essential point of the latter task is to compute the Jacobian for the change of variables from the independent elements of A to the eigenvalues and the independent variables associated with the eigenvectors. Also, because only the eigenvalue PDF is being computed, one must integrate out the eigenvector dependence. In fact the dependence in the Jacobian on the eigenvalues separates from the dependence on the eigenvectors, so the task of performing the integration does not become an issue. Explicitly, one finds (see e.g. [21])

1. Introduction

Recently, it has been realized [1; 2] that conformal maps exhibit an integrable structure: conformal maps of compact simply connected domains bounded by analytic curves provide a solution to the dispersionless limit of the two-dimensional Toda hierarchy. As is well known from the theory of solitons, solutions of an integrable hierarchy are represented by τ-functions. The dispersionless limit of the τ-function emerges as a natural object associated with the curves. In this paper we discuss the τ-function for simple analytic curves and its connection to the inverse potential problem, area preserving diffeomorphisms, the Dirichlet boundary problem, and matrix models.

2. The Inverse Potential Problem

Define a closed analytic curve as a curve that can be parametrized by a function z = x + iy = z(w), analytic in a domain that includes the unit circle |w| = 1. Consider a closed analytic curve 7 in the complex plane and denote by D+ and D- the interior and exterior domains with respect to the curve. The point z = 0 is assumed to be in D+. Assume that the domain D+ is filled homogeneously with electric charge, with a density that we set to 1.

1. Phase Transitions in Invariant Hermitian Ensembles

Random matrix ensembles have been extensively studied for several decades, since the early works of E. Wigner and F. Dyson, as effective mathematical reference models for the descriptions of statistical properties of the spectra of complex physical systems. In the past twenty years new applications spurned a large literature both in theoretical physics and among mathematicians. Several monographs review different sides of the physics literature of the past few decades, such as [7; 10; 18; 40; 41; 59; 92; 94]. Their combined bibliography, although very incomplete, exceeds a thousand papers. Sets of lecture notes are [102; 58; 5; 34; 85; 66]. The classic reference is Mehta's book [82].

For a long time studies and applications of random matrix theory in large part were limited to the choice of gaussian random variables for the independent entries of the random matrix. This was due both to the dominant role of the normal distribution in probability theory as well as to the nice analytic results which were obtained. Increasingly, in the past two decades, a wide variety of matrix ensembles were considered, where the joint probability distribution for the random entries depends on a number of parameters.

1. Introduction

Our first aim of this article is to convince the reader that matrix integrals, exactly calculable or not, can always be interpreted in some sort of combinatorial way as generating functions for decorated graphs of given genus, with possibly specified vertex and/or face valencies. We show this by expressing pictorially the processes involved in computing Gaussian integrals over matrices, what physicists call generically Feynman rules. These matrix diagrammatic techniques have been first developed in the context of quantum chromodynamics in the limit of large number of colors (the size of the matrix) [1; 2], and more recently in the context of two-dimensional quantum gravity, namely the coupling of two-dimensional statistical models (matter theories) to the fluctuations of the two-dimensional space into surfaces of arbitrary topologies (gravity) [3]. These toy models for noncritical string theory are a nice testing ground for physical ideas, and have led to many confirmations of continuum field-theoretical results in quantum gravity.

Introduction

In last few years there appeared a number of papers indicating a strong connection of certain asymptotic problems of enumerative combinatorics and representation theory of symmetric groups with the random matrix theory; see [Baik et al. 1999a; 1999b; Baik and Rains 1999a; 1999b; Borodin 1998a; 1998b; 1999; ≥ 2001; Borodin and Olshanski 1998a; 1998b; 2000a; Borodin et al. 2000; Johansson 2000; 1999; Okounkov 1999b; 1999a; Olshanski 1998a; 1998b; Tracy and Widom 1998; 1999], for a partial list. Such a connection was also anticipated in earlier works [Regev 1981; Kerov 1993; 1994]. For other interesting connections see also [Borodin 2000b; Borodin and Okounkov 2000; Okounkov 2001].

In this paper we suggest a hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with so-called “β = 2 ensembles” arising in random matrix theory. (These ensembles are characterized by the property that their correlation functions have determinantal form with a scalar kernel; see below.) We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group; see [Kerov et al. 1993] and Section 3 below.

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.