Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

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[-βθ(0) N^2] where the Dyson index βcharacterizes the ensemble and the exponent θ(0)=(\ln 3)/4=0.274653... is universal. We compute the probability that the eigenvalues lie in the interval [ζ_1,ζ_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 [ζ_1,ζ_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.