Random matrices: tail bounds for gaps between eigenvalues

Abstract

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications. Mathematics Subject Classification

Matrix regularizing effects of Gaussian perturbations

“…Upper bounds on the probability of two close eigenvalues were proved by Nguyen, Tao and Vu [22]. Recently, universality of local eigenvalue statistics for deformed Wigner ensembles was studied by O'Rourke and Vu [23], Knowles and Yin [15, Section 12] and Lee, Schnelli, Stetler, and Yau [16].…”

Convergence of Local Statistics of Dyson Brownian Motion

“…We remark that a form of level repulsion was obtained in [24,27] for Wigner ensembles under no smoothness assumptions on the matrix elements. However, the estimates obtained are not strong enough for our methods.…”

“…Weaker level repulsion estimates but with no smoothness condition were obtained in [24,27]. The proof of [11] was modified in [6] to include the case when the Wigner ensemble is not smooth but instead is the sum of a (possibly non-smooth) Wigner matrix and an independent Gaussian part.…”

On a Conjecture of Godsil Concerning Controllable Random Graphs

“…Specifically, we will exploit advances in Littlewood-Offord theory recently developed by Rudelson and Vershynin [24,26,31]. The authors' previous work [20] also used a stochastic approach to study controllability properties of random systems and was based on recent advances by Tao and Vu [28] and Nguyen, Tao, and Vu [18] concerning gaps between eigenvalues of random matrices. There it is shown that the controllability and minimal controllability of systems is a generic property, even for systems of a very discrete nature.…”