The singularity probability of a random symmetric matrix is exponentially small

TL;DR

This paper proves that the probability of a random symmetric ±1 matrix being singular decreases exponentially with size, confirming a longstanding conjecture in random matrix theory.

Contribution

It establishes an exponential bound on the singularity probability of random symmetric ±1 matrices, resolving a well-known conjecture.

Findings

Probability of singularity is at most e^{-cn} for some constant c>0

Confirms the exponential decay of singularity probability in symmetric ±1 matrices

Resolves a long-standing open problem in random matrix theory

Abstract

Let $A$ be drawn uniformly at random from the set of all $n\times n$ symmetric matrices with entries in $\{-1,1\}$. We show that \[ \mathbb{P}( \det(A) = 0 ) \leq e^{-cn},\] where $c>0$ is an absolute constant, thereby resolving a well-known conjecture.