Singularity of random symmetric matrices -- a combinatorial approach to improved bounds

TL;DR

This paper improves bounds on the probability that a random symmetric Bernoulli matrix is singular, using a new combinatorial approach, showing it decays faster than previous estimates for large matrices.

Contribution

It introduces an enhanced combinatorial method to bound the singularity probability of symmetric Bernoulli matrices, achieving significantly tighter bounds than prior work.

Findings

Singularity probability bound: at most 2^{-n^{1/4}√log n / 1000} for large n

Improved upon previous polynomial and exponential bounds

Utilizes an extended combinatorial approach to discrete random matrix theory

Abstract

Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_n$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_n$, due to Vershynin (2011), is $2^{-n^c}$, for some unspecified small constant $c > 0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_n$ is at most $2^{-n^{1/4}\sqrt{\log{n}}/1000}$ for all sufficiently large $n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.