On the least singular value of random symmetric matrices

Let Fn be an n by n symmetric matrix whose entries are bounded by n γ for some γ > 0. Consider a randomly perturbed matrix Mn = Fn +Xn, where Xn is a random symmetric matrix whose upper diagonal entries xij , 1 ≤ i ≤ j, are iid copies of a random variable ξ. Under a very general assumption on ξ, we show that for any B > 0 there exists A > 0 such that P(σn(Mn) ≤ n−A) ≤ n−B .