Invertibility of symmetric random matrices1. Issue 2 (11th May 2012)

• Record Type: Journal Article
• Authors: Vershynin, Roman
• Is Part Of: Random structures & algorithms. Volume 44:Issue 2(2014)
• Page Start: 135
• Page End: 182
• Publication Date: 2012-05-11
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20429
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We study <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w7cg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20429:rsa20429-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives> symmetric random matrices <italic>H</italic>, possibly discrete, with iid above‐diagonal entries. We show that <italic>H</italic> 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" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20429:rsa20429-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>exp</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>, and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg40d2w755" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20429:rsa20429-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>. Furthermore, the spectrum of <italic>H</italic> 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" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20429:rsa20429-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></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</p> </abstract>