1. Introduction

We present an overview of some recent developments in the application of random matrix analysis to the topological combinatorics of surfaces. Such applications have a long history about which we should say a few words at the outset. The combinatorial objects of interest here are maps. A map is an embedding of a graph into a compact, oriented and connected surface X with the requirement that the complement of the graph in X should be a disjoint union of simply connected open sets. If the genus of X is g, this object is referred to as a g-map. The notion of g-maps was introduced by Tutte [1968] and his collaborators in the 1960s as part of their investigations of the four color conjecture.

In the early 1980s Bessis, Itzykson and Zuber, a group of physicists studying 't Hooft's diagrammatic approaches to large-N expansions in quantum field theory, discovered a profound connection between the problem of enumerating g-maps and random matrix theory [Bessis et al. 1980]. That seminal work was the basis for bringing asymptotic analytical methods into the study of maps and other related combinatorial problems. Subsequently other physicists [Douglas and Shenker 1990; Gross and Migdal 1990] realized that the matrix model diagrammatics described by Bessis et al. provide a natural means for discretizing the Einstein–Hilbert action in two dimensions.

1. Introduction

The one-dimensional Kardar–Parisi–Zhang (KPZ) equation,

is a well known prototypical equation which describes a growing interface [Kardar et al. 1986; Barabási and Stanley 1995]. Here h(x, t) represents the height of the surface at position x ∈ ℝ and time t ≥ 0. The first term represents a non linearity effect and the second term describes a smoothing mechanism. The parameters λ and υ measure the strengths of these effects. The last term η(x, t) indicates the existence of randomness in our description of surface growth. For the standard KPZ equation it is taken to be the Gaussian white noise with covariance,

Here and in the remainder of the article, (...) indicates an average with respect to the randomness η.

The KPZ equation (1) is a nonlinear stochastic partial differential equation (SPDE), which is difficult to handle in general. But fortunately the KPZ equation has a nice integrable structure which has allowed detailed studies of its properties. In particular the one-point height distribution has been computed explicitly for three different initial conditions: the narrow wedge h(x,0) = -|x|/δ,δ→0

[Sasamoto and Spohn 2010a; Amir et al. 2011], the flath(x,0)=0,x∈ ℝ [Calabrese and Le Doussal 2011], and the stationary, h(x, 0) = B(x) [Imamura and Sasamoto 2012], cases. Here B(x),x∈ ℝ represents the (two-sided) onedimensional Brownian motion with B(0)= 0. From the narrow wedge initial condition the surface grows to a shape of parabola macroscopically, which is representative of a curved surface. The flat case has been the most typical initial condition for Monte Carlo simulation studies, whereas the stationary case is regarded as one of the most important situation from the point of view of nonequilibrium statistical mechanics

1. Harish-Chandra formula, Duistermaat–Heckman measure and Gelfand–Tsetlin patterns

DefineJλ(x) = h(λ)-1 det(eλi xj) For each x, Jλ(x) is an analytic function of λ; in particular,

The functions Jλ(x) play a central role in random matrix theory. For example, if

Λ and X are Hermitian matrices with eigenvalues given by λ and x, respectively, Then

where the integral is with respect to normalised Haar measure on the unitary

group. This is known as the Harish-Chandra, or Itzykson–Zuber, formula.

1.Introduction

The first decade of the 21st century and a couple of preceding years have been marked by a series of remarkable analytical developments, which have revealed profound and rigorous connections among random matrix theory, combinatorial problems, and the physical problem of the fluctuating interface growth ([Baik and Rains 2001; Kriecherbauer and Krug 2010; Sasamoto and Spohn 2010a; Corwin 2012] and references therein). Their primary conclusions in terms of the interface growth problem, first pointed out by Johansson [2000] for TASEP and by Prähofer and Spohn [2000] for the PNG model, are the following [Kriecherbauer and Krug 2010; Sasamoto and Spohn 2010a; Corwin 2012]: (i) The distribution function and the spatial correlation function were obtained rigorously for the asymptotic interface fluctuations. (ii) The results depend on the global shape of the interfaces, or on the initial condition. For the two prototypical cases of the curved and flat growing interfaces, the distribution function is given by the Tracy–Widom distribution [Tracy and Widom 1994; 1996] for GUE and GOE, respectively, and the spatial two-point correlation function by the covariance of the Airy2 and Airy1 process [Prähofer and Spohn 2002; Sasamoto 2005],

respectively. (iii) All or some of these conclusions were reached for a number of models, namely the TASEP [Johansson 2000; Borodin et al. 2007; 2008; Sasamoto 2005] and the PASEP [Tracy and Widom 2009], the PNG model [Prähofer and Spohn 2000; 2002; Borodin et al. 2008], and the KPZ equation [Sasamoto and Spohn 2010c; 2010b; Amir et al. 2011; Calabrese et al. 2010; Dotsenko 2010; Calabrese and Le Doussal 2011; Prolhac and Spohn 2011].

This is much harder. I quote from Feynman [1964] at some length

With this chapter we begin a new subject which will occupy us for some time. It is the first part of the analysis of the properties of matter from the physical point of view, in which, recognizing that matter is made out of a great many atoms, or elementary parts, which interact electrically and obey the laws of mechanics, we try to understand why various aggregates of atoms behave the way they do.

It is obvious that this is a difficult subject, and we emphasize at the beginning that it is in fact an extremely difficult subject, and that we have to deal with it differently than we have dealt with the other subjects so far. In the case of mechanics and in the case of light, we were able to begin with a precise statement of some laws, like Newton's laws, or the formula for the field produced by an accelerating charge, from which a whole host of phenomena could be essentially understood, and which would produce a basis for our understanding of mechanics and light from that time on. That is, we may learn more later, but we do not learn different physics, we only learn better methods of mathematical analysis to deal with the situation.

This subject is not so classical as what's gone before. It has its roots in practical statistics (1930 or so), quantum mechanics (1958), and, very recently, in an astonishing variety of other questions, touched upon below. §12.7 et seq. are technically more demanding than any thing we've done before, employing for the most part radically new methods, but it's fascinating and worth the effort. Mehta [1967] is the best general reference.

The Gaussian orthogonal ensemble (GOE)

Quantum-mechanically speaking, the energy levels E of a large collection of n like atoms, without internal degrees of freedom such as angular momentum or spin, may be found by solving Hψ ≡ −Δψ + Vψ = Eψ, in which Δ is Laplace's operator in ℝ3n = ℝ3 × … × ℝ3, one copy for each atom to record its location, and V is the (classical) total potential energy of such a configuration. But, if n is big, the computation is already way out of reach of the biggest machine. What to do? Well, you can stop there, or you can look for something simpler, something you can compute, to see if that helps. Wigner [1958, 1967] proposed the simplest such caricature in connection with the scattering of neutrinos off heavy nuclei, replacing H by a typical, real symmetric, n × n matrix x = (xij :1 ≤ i, j ≤ n) and looking at its spectrum as n ↑ ∞ far away from Nature perhaps, but never mind.