1. Introduction

Using random matrices to count combinatorial objects is not a new idea. It stems from the pioneering work [Brezin et al. 1978], which showed how the perturbative expansion of a simple nongaussian matrix integral led, using standard Feynman diagram techniques, to the counting of discretized surfaces. It has resulted in many applications: from the physical side, it allowed to define a discretized version of 2D quantum gravity [Di Francesco et al. 1995] and to study various statistical models on random lattices [Kazakov 1986; Rostov 1989; Gaudin and Rostov 1989; Rostov and Staudacher 1992]. From the mathematical side, let us cite the Rontsevitch integral [Rontsevich 1991; Witten 1991; Itzykson and Zuber 1992], and the counting of meanders and foldings [Makeenko 1996; Di Francesco et al. 1997; 1998].

Here we shall try to apply this idea to the field of knot theory. Our basic aim will be to count knots or related objects. The next section defines these objects, and is followed by a brief overview of matrix models and how they can be related to knots.

1. Introduction

This is a survey article dealing with connections between orthogonal polynomials, functional equations they satisfy, and some extremal problems. Although the results surveyed are not new we believe that we are putting together results from different sources which appear together for the first time, many of them are of recent vintage. One question in the theory of orthogonal polynomials is how the zeros of a parameter dependent sequence of orthogonal polynomials change with the parameters involved. Stieltjes [1885a; 1885b] proved that the zeros of Jacobi polynomials increase with β and decrease with a for a > -1 and β > -1. The Jacobi polynomials satisfy the following orthogonality relation [Szego 1975, (4.3.3)]:

In Section 2 we state Stieltjes's results and describe the circle of ideas around them. Section 3 surveys the Coulomb Fluid method of Freeman Dyson and its potential theoretic set-up. We shall use the shifted factorial notion [Andrews et al. 1999]

1. Introduction

A class of problems — important for their applications to computer science and computational biology as well as for their inherent mathematical interest — is the statistical analysis of a string of random symbols. The symbols, called letters, are assumed to belong to an alphabet A of fixed size k. The set of all such strings (or words) of length N, W(A,N), forms the sample space in the statistical analysis of these strings. A natural measure on W is to assign each letter equal probability, namely 1/ k, and to define the probability measure on words by the product measure. Thus each letter in a word occurs independently and with equal probability. We call such random word models homogeneous. Of course, for some applications, each letter in the alphabet does not occur with the same frequency and it is therefore natural to assign to each letter i a probability pi. If we again use the product measure for the words (letters in a word occur independently), then the resulting random word models are called inhomogeneous. Fixing an ordering of the alphabet A, a weakly increasing subsequence of a Word.

1. Introduction la. Isomonodromic Deformation Equations. We consider rational covariant derivative operators on the punctured Riemann sphere, having the form They have regular singular points at and an irregular singularity at with Poincare index 1. If the residue matrices are deformed differentiably with respect to the parameters and the monodromy (including Stokes parameters and connection matrices) of the operator will be invariant under such deformations, as was shown in [Jimbo et al. 1980; Jimbo et al. 1981], provided the differential equations implied by the commutativity conditions.

1. Introduction

A well-known theme in random matrix theory (RMT), zeta functions, quantum chaos, and statistical mechanics, is the universality of scaling limits of correlation functions. In RMT, the relevant correlation functions are for eigenvalues of random matrices (see [De; TW; BZ; BK; So] and their references). In the case of zeta functions, the correlations are between the zeros [KS]. In quantum dynamics, they are between eigenvalues of ‘typical’ quantum maps whose underlying classical maps have a specified dynamics. In the ‘chaotic case’ it is conjectured that the correlations should belong to the universality class of RMT, while in integrable cases they should belong to that of Poisson processes. The latter has been confirmed for certain families of integrable quantum maps, scattering matrices and Hamiltonians (see [Ze2; RS; Sa; ZZ] and their references). In statistical mechanics, there is a large literature on universality of critical exponents [Car]; other rigorous results include analysis of universal scaling limits of Gibbs measures at critical points [Sin]. In this article we are concerned with a somewhat new arena for scaling and universality, namely that of RPT (random polynomial theory) and its algebro-geometric generalizations [Han; Hal; BBL; BD; BSZ1; BSZ2; SZ1; NV]. The focus of these articles is on the configurations and correlations of zeros of random polynomials and their generalizations, which we discuss below. Random polynomials can also be used to define random holomorphic maps to projective space, but we leave that for the future. Our purpose here is partly to review the results of [SZ1; BSZ1; BSZ2] on universality of scaling limits of correlations between zeros of random holomorphic sections on complex manifolds.

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.