Random symmetric matrices are almost surely nonsingular

Abstract

Let $Q_n$ denote a random symmetric ($n\times n)$-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values $0$ and $1$ with probability $1/2$). We prove that $Q_n$ is nonsingular with probability $1-O(n^{-1/8+\delta })$ for any fixed $\delta >0$. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices