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2014 Largest eigenvalues and eigenvectors of band or sparse random matrices
Florent Benaych-Georges, Sandrine Péché
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Electron. Commun. Probab. 19: 1-9 (2014). DOI: 10.1214/ECP.v19-3027


In this text, we consider an $N$ by $N$ random matrix $X$ such that all but $o(N)$ rows of $X$ have $W$ non identically zero entries, the other rows having less than $W$ entries (such as, for example, standard or cyclic band matrices). We always suppose that $1 \ll W \ll N$. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and $W \gg (\log N)^{6(1+\alpha)}$, where $\alpha$ is a positive parameter depending on the distribution of the entries, then the largest eigenvalue of $X/\sqrt{W}$ converges to the upper bound of its limit spectral distribution, that is $2$, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on $W$, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of $X$, precisely that most of them cannot be essentially localized on less than $W/\log(N)$ entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that the localization length in the edge is $\ll W^{7/5}$ or there is strong interaction between two eigenvectors in an interval oflength $W^{7/5}$.</p>


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Florent Benaych-Georges. Sandrine Péché. "Largest eigenvalues and eigenvectors of band or sparse random matrices." Electron. Commun. Probab. 19 1 - 9, 2014.


Accepted: 30 January 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1297.15038
MathSciNet: MR3164751
Digital Object Identifier: 10.1214/ECP.v19-3027

Primary: 15A52
Secondary: 60F05

Keywords: Band matrices , largest eigenvalues , Localization , random matrices


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