Inverse Littlewood-Offord problems and the singularity of random symmetric matrices (Q410097)

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 \(1\over 2\)). Improving the earlier result by \textit{K. P. Costello, T. Tao} and \textit{V. Vu} [Duke Math. J. 135, No. 2, 395--413 (2006; Zbl 1110.15020)], the author shows that \(M_n\) is nonsingular 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.