3.14 Random Matrices

Organizers J. Baik (Ann Arbor), J. Harnad (Montréal)

3.14.1Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions

Pavel M. Bleher

Indiana University Purdue University Indianapolis bleher@math.iupui.edu

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite N by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an N × N Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expresses in terms of the partition function of a random matrix model with non-polynomial interaction. We use this observation to obtain the large N asymptotic of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the six-vertex model. We compare the obtained exactsolution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn-Justin that the partition function of the sic-vertex model with DWBC has the asymptotics, ZN CN κ eN 2f as N → ∞, and we find the exact value of the exponent κ.

3.14.2Probabilities of a Large Gap in the Scaled Spectrum of Random Matrices

Igor Krasovsky

Brunel Univ. mastiik2@brunel.ac.uk

In the Gaussian Unitary Ensemble of random matrices, the probability of an interval (gap) without eigenvalues in the spectrum (rescaled in a standard way) is given by the sine-kernel Fredholm determinant in the bulk of the spectrum, and by the Airy-kernel Fredholm determinant at the edge. We calculate asymptotics of these determinants for a large gap proving the conjectures of Dyson, Tracy and Widom about the multiplicative constant in these asymptotics. Our method uses analysis of a Riemann-Hilbert problem and can be adapted to calculate such constants in asymptotics of a number of similar determinantal correlation functions of random

matrix theory and exactly solvable models. The talk is based mostly on joint works with P. Deift, A. Its, and X. Zhou.

3.14.3 Random Matrices, Asymptotic Analysis, and d-bar Problems

Kenneth McLaughlin

University of Arizona mcl@math.arizona.edu

The aim of the research is to study random matrix models for which the external field is outside the analytic class.

3.14.4 Central Limit Theorems for Non-intersecting Random Walks

Toufic Suidan

UC Santa Cruz tsuidan@ucsc.edu

We describe several central limit theorems for non-intersecting random walks. The limiting distributions which arise are related to classical random matrix theory. Connections to last passage percolation and other models will be discussed. This work is joint with Jinho Baik.

3.14.5On the Distribution of Largest Eigenvalues in Random Matrix Ensembles

Alexander Soshnikov

UC Davis soshniko@math.ucdavis.edu

In the talk, we will consider the Wigner and Wishart ensembles of random matrices and their generalizations. We will discuss the spectral properties of random matrices from these ensembles, in particular the distribution of the largest (smallest) eigenvalues.

3.14.6 Non-Intersecting Brownian Excursions

Craig A. Tracy

UC Davis tracy@math.ucdavis.edu

A Brownian excursion is a Brownian path starting at the origin at time t = 0 and ending at the origin at time t = 1 and conditioned to remain positive for 0 < t < 1.