Extreme value statistics of eigenvalues of Gaussian random matrices

TLDR

The authors derive exact asymptotic expressions for the probability of large deviations of the largest (or smallest) eigenvalue in Gaussian orthogonal, unitary, and symplectic ensembles, and compute the probability that eigenvalues lie in an interval \([ζ_1,ζ_2]\), enabling the joint distribution of the minimum and maximum eigenvalues. They find that the probability all eigenvalues are positive (or negative) decays as \(\exp[-βθ(0)N^2]\) with \(θ(0)=(\ln 3)/4≈0.27465\), and that the average density of states for eigenvalues restricted to \([ζ_1,ζ_2]\) generalizes the Wigner semicircle law, exhibiting an inverse‑square‑root singularity at the barriers, with all results confirmed by numerical simulations. These results were first announced in Phys.

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 approximately exp [-beta theta(0)N2] 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 [zeta1,zeta2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a by-product, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [zeta1,zeta2] , 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. Some of the results presented in detail here were announced in a previous paper [D. S. Dean and S. N. Majumdar, Phys. Rev. Lett. 97, 160201 (2006)].