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Largest eigenvalues and eigenvectors of band or sparse random matrices


 
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1. Title Title of document Largest eigenvalues and eigenvectors of band or sparse random matrices
 
2. Creator Author's name, affiliation, country Florent Benaych-Georges; Université Paris Descartes; France
 
2. Creator Author's name, affiliation, country Sandrine Péché; Université Paris Diderot
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random matrices ; band matrices ; largest eigenvalues ; localization
 
3. Subject Subject classification 15A52; 60F05
 
4. Description Abstract

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 $W^{7/5}$.
 
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7. Date (YYYY-MM-DD) 2014-01-30
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3027
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3027
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
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