Singularity of random symmetric matrices revisited

TL;DR

This paper improves the upper bound on the probability that a random symmetric ±1 matrix is singular, providing a simpler method that advances understanding of matrix singularity probabilities.

Contribution

It introduces a new, simpler approach that tightens the upper bound on singularity probability of symmetric ±1 matrices, surpassing previous results.

Findings

Upper bound on singularity probability improved to exp(-c(n log n)^{1/2})

Method is simpler and different from previous approaches

Advances theoretical understanding of symmetric random matrix singularity

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.