Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

TL;DR

This paper proves that random symmetric Bernoulli matrices are non-singular with high probability, using inverse Littlewood-Offord results for quadratic forms, advancing understanding of matrix invertibility.

Contribution

It introduces an improved bound on the probability of singularity for random symmetric Bernoulli matrices using novel inverse Littlewood-Offord techniques.

Findings

Random symmetric Bernoulli matrices are non-singular with high probability

The probability of singularity is at most O(n^{-C}) for any positive C

Inverse Littlewood-Offord results for quadratic forms are developed

Abstract

Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.