Inverse Littlewood–Offord problems and the singularity of random symmetric matrices

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 value $-1$ and $1$ with probability $1/2$). Improving the earlier result by Costello, Tao, and Vu [4], we show that $M_n$ is nonsingular with probability $1-O\left(n^{-C}\right)$ for any positive constant $C$. The proof uses an inverse Littlewood–Offord result for quadratic forms, which is of interest of its own.