Singularity of random symmetric matrices revisited

Abstract

Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n\times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp (-c(n\log n{)}^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp (-c{n}^{1/2})$ on the singularity probability, our method is different and considerably simpler.

12 pages