Large Deviations of Extreme Eigenvalues of Random Matrices

TL;DR

This paper analytically derives the probabilities of large deviations for the extreme eigenvalues of Gaussian random matrices and generalizes the Wigner semi-circle law for constrained eigenvalue densities.

Contribution

It provides exact calculations of large deviation probabilities and extends the Wigner semi-circle law to matrices with eigenvalues above a fixed threshold.

Findings

Probability of all eigenvalues positive/negative decreases as exp[-βθ(0)N^2]

Universal exponent θ(0) = (ln 3)/4 for large N

Density of states shows inverse square-root singularity at threshold ζ

Abstract

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N\times N) random matrix are positive (negative) decreases for large N as \exp[-\beta \theta(0) N^2] where the parameter \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number \zeta, thus generalizing the celebrated Wigner semi-circle law. The density of states generically exhibits an inverse square-root singularity at \zeta.