Singularity of random symmetric matrices -- simple proof

TL;DR

This paper presents a straightforward and concise proof providing an upper bound on the probability that a random symmetric matrix with ±1 entries is singular, advancing understanding of matrix invertibility in probabilistic combinatorics.

Contribution

It introduces a simple, self-contained proof for a significant upper bound on the singularity probability of random symmetric ±1 matrices, simplifying previous approaches.

Findings

Established a new upper bound on singularity probability

Provided a simple, self-contained proof

Enhanced understanding of symmetric random matrix invertibility

Abstract

In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random $\pm 1$ symmetric matrix is singular.