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Invertibility of symmetric random matrices
Author(s) -
Vershynin Roman
Publication year - 2014
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20429
Subject(s) - diagonal , mathematics , delocalized electron , spectrum (functional analysis) , combinatorics , random matrix , struct , polynomial , singularity , constant (computer programming) , discrete mathematics , pure mathematics , computer science , physics , quantum mechanics , mathematical analysis , eigenvalues and eigenvectors , geometry , programming language
Abstract We study n × 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 | | H − 1 | | = O ( 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 Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 2014