Large Deviations for Random Matrices

TL;DR

This paper establishes a large deviation principle for the eigenvalues of large symmetric random matrices with i.i.d. entries, linking the spectrum's behavior to Hilbert-Schmidt kernels and deriving a rate function based on the entries' distribution.

Contribution

It introduces a large deviation framework for the spectrum of random matrices, connecting spectral properties to kernel functions and providing explicit rate functions.

Findings

Eigenvalues of order n occur with exponentially small probability.

Spectral limits correspond to Hilbert-Schmidt kernels on [0,1]^2.

Rate function derived from the Cramer rate function of entry distribution.

Abstract

We prove a large deviation result for a random symmetric n x n matrix with independent identically distributed entries to have a few eigenvalues of size n. If the spectrum S survives when the matrix is rescaled by a factor of n, it can only be the eigenvalues of a Hilbert-Schmidt kernel k(x,y) on [0,1] x [0,1]. The rate function for k is $I(k)=1/2\int h(k(x,y) dxdy$ where h is the Cramer rate function for the common distribution of the entries that is assumed to have a tail decaying faster than any Gaussian. The large deviation for S is then obtained by contraction.