Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

TL;DR

This paper derives exact asymptotic probabilities for large deviations of eigenvalues in Gaussian random matrices, revealing universal behaviors and extending classical laws like Wigner's semi-circle to constrained ensembles.

Contribution

It provides the first exact asymptotic formulas for large deviations of eigenvalues in Gaussian ensembles, including universal probability decay rates and density of states in restricted intervals.

Findings

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

Derived the joint distribution of minimum and maximum eigenvalues

Extended Wigner semi-circle law to eigenvalues restricted in intervals

Abstract

We compute exact asymptotic results for the probability of the occurrence of large deviations 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 (NxN) random matrix are positive (negative) decreases for large N as ~\exp[-\beta \theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a byproduct, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner semi-circle law to these restricted ensembles. It is found that the density of states generically exhibits an inverse square-root singularity at the location of the barriers. These results are confirmed by numerical simulations.