Invertibility of symmetric random matrices1. Issue 2 (11th May 2012) - Vershynin, Roman
Abstract We study <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w7cg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />n×n</alternatives> symmetric random matrices H, possibly discrete, with iid above‐diagonal entries. We show that H is singular with probability at most <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w7d1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />exp(−nc)</alternatives>, and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w755" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />||H−1||=O(n)</alternatives>. Furthermore, the spectrum of H is delocalized on the optimal scale <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w78t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />o(n−1/2)</alternatives>. 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 Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 2014