# On the singularity of random symmetric matrices

## TL;DR

This paper improves the upper bound on the probability that a random symmetric $ ext{-}1$ matrix is singular, using an inverse Littlewood-Offord theorem inspired by hypergraph container methods.

## Contribution

It introduces a new inverse Littlewood-Offord theorem in $ ext{Z}_p^n$ and significantly tightens the bound on singularity probability of random symmetric matrices.

## Key findings

- Probability of singularity is at most exp(-Omega(sqrt(n)))
- Improves previous bound of exp(-Omega(n^{1/4} sqrt(log n)) )
- Introduces a new inverse Littlewood-Offord theorem

## Abstract

A well-known conjecture states that a random symmetric $n \times n$ matrix with entries in $\{-1,1\}$ is singular with probability $\Theta\big( n^2 2^{-n} \big)$. In this paper we prove that the probability of this event is at most $\exp\big( - \Omega( \sqrt{n} ) \big)$, improving the best known bound of $\exp\big( - \Omega( n^{1/4} \sqrt{\log n} ) \big)$, 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.