Recent progress in combinatorial random matrix theory

Abstract

We are going to discuss recent progress on many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

The Sixth Moment of Random Determinants

“…For a recent survey of results on random matrices, see [16]. In this paper, we extend the first line of work by analyzing the sixth moment of the determinant of a random matrix.…”

Quantitative invertibility of non-Hermitian random matrices

“…In last few years there has been a significant progress in this research direction (also, as corollaries of quantitative results), in particular, the problem of estimating the sigularity probability of adjacency matrices of random regular (di)graphs [37,61,64], of Bernoulli random matrices [91,56,42] and, more generally, discrete matrices with i.i.d entries [43], of random symmetric matrices [13,29,11,12]. We refer to a recent survey [96] for a discussion and further references.…”

The singularity probability of a random symmetric matrix is exponentially small

“…Again, it is generally believed that P(det A = 0) = Θ(n 2 2 −n ) (see, e.g. [8,52,53]) but progress has come more slowly. The problem of showing that A is almost surely non-singular goes back, at least, to Weiss in the early 1990s and was not resolved until 2005 by Costello, Tao and Vu [8], who obtained the bound of P(det(A) = 0) n −1/8+o (1) .…”