Invertibility of symmetric random matrices
Probability
2014-03-05 v4 Functional Analysis
Abstract
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.
Cite
@article{arxiv.1102.0300,
title = {Invertibility of symmetric random matrices},
author = {Roman Vershynin},
journal= {arXiv preprint arXiv:1102.0300},
year = {2014}
}
Comments
53 pages. Minor corrections, changes in presentation. To appear in Random Structures and Algorithms