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Introduction to Random Matrices

These notes provide an introduction to the theory of random matrices. The central quantity studied is $τ(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/π} {\sinπ(x-y)\over x-y} χ_I(y)$. Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $χ_I(y)$ 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 $τ(a)$. Also $τ(a)$ 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{ô}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{é} 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 $s$.