  We study n by n symmetric random matrices H, possibly discrete, with iid
above-diagonal entries. We show that H is singular with probability at most
exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}).
Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}).
These results improve upon a polynomial singularity bound due to Costello, Tao
and Vu, and they generalize, up to constant factors, results of Tao and Vu, and
Erdos, Schlein and Yau.
