Introduction to Random Matrices

TL;DR

This paper introduces the theory of random matrices, focusing on the probability of eigenvalue gaps in GUE, and presents new derivations of related integrable equations and asymptotic formulas.

Contribution

It provides a simplified derivation of nonlinear integrable equations related to random matrices and connects these to Painlevé V equations for single intervals.

Findings

Derived a new simplified system of nonlinear equations for eigenvalue distributions.

Connected the eigenvalue gap probabilities to Painlevé V equations.

Provided asymptotic formulas for eigenvalue probabilities in large intervals.

Abstract

These notes provide an introduction to the theory of random matrices. The central quantity studied is $\tau(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y)$. Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $\chi_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in $I$ is equal to $\tau(a)$. Also $\tau(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$'s are the independent variables) that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{\'e} V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$.