Large Deviations for Random Matrices

2011/06/22 by Chatterjee, Sourav, Varadhan, S. R. S.
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1106.4366

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∫ 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.

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