On the singularity of random symmetric matrices

Abstract

A well-known conjecture states that a random symmetric $n\times n$ matrix with entries in $\{-1,1\}$ is singular with probability $\mathrm{\Theta }(n^2{2}^{-n})$. We prove that the probability of this event is at most $exp(-\mathrm{\Omega }(\sqrt{n}))$, improving the best-known bound of $exp(-\mathrm{\Omega }(n^{1∕4}\sqrt{logn}))$, which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood–Offord theorem in ${\mathbb{Z}}_p^n$ that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers.