Random weighted projections, random quadratic forms and random eigenvectors. Issue 4 (2nd July 2014)

• Record Type: Journal Article
• Authors: Vu, Van; Wang, Ke
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 792
• Page End: 821
• Publication Date: 2014-07-02
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20561
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms. (2) The infinity norm of most unit eigenvectors of a random ±1 matrix is of order <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvkhd4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20561:rsa20561-math-0001" 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:msqrt><mml:mrow><mml:mi>log</mml:mi><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. (3) An estimate on the threshold for the local semi‐circle law which is tight up to a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvkh0x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20561:rsa20561-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mrow><mml:mi>log</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula> factor. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 792–821, 2015</p> </abstract>