On the least singular value of random symmetric matrices

Let F-n be an n by n symmetric matrix whose entries are bounded by n(gamma) for some gamma > 0. Consider a randomly perturbed matrix M-n = F-n + X-n, where X-n is a random symmetric matrix whose upper diagonal entries x(ij), 1 <= i <= j, are iid copies of a random variable xi. Under a very general assumption on xi, we show that for any B > 0 there exists A > 0 such that P (sigma(n) (M-n) <= n(-A)) <= n(-B).