This is an announcement for the paper "Invertibility of symmetric random matrices" by Roman Vershynin.
Abstract: Let H be an n by n symmetric random matrix whose above-diagonal entries are general iid random variables (possibly discrete) with zero mean, unit variance, and subgaussian tail decay. We prove that H is singular with probability at most exp(n^{-c}) for some constant c>0, and that the spectral norm of the inverse of H is O(\sqrt{n}) with high probability. More generally, the spectrum of H is delocalized -- with high probability, there are no eigenvalues in an arbitrary fixed interval of the optimal length o(n^{-1/2}). The delocalization result also holds under the fourth moment assumption on the entries of H. These results improve upon the polynomial singularity bound O(n^{-1/8+epsilon}) due to Costello, Tao and Vu, and they generalize, up to constant factors, previous results for distributions whose first few moments match the moments of the normal distribution (due to the universality results of Tao and Vu) and for continuous distributions in the bulk of the spectrum (due to Erd"os, Schlein and Yau).
Archive classification: math.PR math.FA
Mathematics Subject Classification: 15B52
Remarks: 52 pages
Submitted from: romanv@umich.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1102.0300
or