1. Six-vertex model

The six-vertex model, or the model of two-dimensional ice, is stated on a square lattice with arrows on edges. The arrows obey the rule that at every vertex there are two arrows pointing in and two arrows pointing out. This rule is sometimes called the ice-rule. There are only six possible configurations of arrows at each vertex, hence the name of the model; see Figure 1.

We will consider the domain wall boundary conditions (DWBC), in which the arrows on the upper and lower boundaries point into the square, and the ones on the left and right boundaries point out. One possible configuration with DWBC on the 4 × 4 lattice is shown on Figure 2.

The name of the square ice comes from the two-dimensional arrangement of water molecules, H2O, with oxygen atoms at the vertices of the lattice and one hydrogen atom between each pair of adjacent oxygen atoms. We place an arrow in the direction from a hydrogen atom toward an oxygen atom if there is a bond between them. Thus, as we already noticed before, there are two in-bound and two out-bound arrows at each vertex.

Figure 1. The six arrow configurations allowed at a vertex.

Figure 2. An example of a 4 x 4 configuration (left) and the corresponding ice crystal (right).

For each possible vertex state we assign a weight wi, i =1, ...., 6, and define, as usual, the partition function, as a sum over all possible arrow configurations of the product of the vertex weights,

where Vn is the n × n set of vertices,σ (x)∈{1,...,6} is the vertex configuration of σ at vertex x according to Figure 1, and Ni (σ) is the number of vertices of type i in the configuration σ.

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.

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.