Abstract not found

. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1. Introduction Random matrix theory is actively developing. Among numerous topics of the theory and its various applications those related to the asymptotic eigenvalue distribution of random matrices of large order are of considerable interest. An important role in this branch of the theory plays the eigenvalue counting measure defined for any Hermitian or real symmetr...

Monodromy deformation approach to the scaling limit of the Painlevé first equation

We discuss the asymptotics of the Schlesinger and Garnier systems with a large parameter, Boutroux and Whitham equations for modulated spectral curves for asymptotic solutions of isomonodromy problems and properties of the discriminant sets. 1

These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very naturally as integrals over the tangent space to symmetric spaces, as integrals over groups and finally as integrals over symmetric spaces. An important part of these lectures is devoted to showing that these matrix integrals, upon apropriately adding time-parameters, are natural tau-functions for integrable lattices, like the Toda, Pfaff and Toeplitz lattices, but also for integrable PDE’s, like the KdV equation. These matrix integrals or Fredholm determinants also satisfy Virasoro constraints, which combined with the integrable equations lead to (partial) differential equations for the original probabilities.

In this paper we construct a class of random matrix ensembles labelled by a real parameter α ∈ (0,1), whose eigenvalue density near zero behaves like |x | α. The eigenvalue spacing near zero scales like 1/N 1/(1+α) and thus these ensembles are representatives of a continous series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the former case we obtain formulas for the eigenvalue density, while in the latter case we obtain approximate expressions for the scaling functions in the microscopic limit using a very simple approximate method based on the location of zeroes of orthogonal polynomials. 1