Largest eigenvalues and eigenvectors of band or sparse matrices

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 lass than $W$ entries (such as, for example, standard or cyclic band matrices). We always suppose that 1 << W << N. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and W >> (\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 either the localization length in the edge is << W^{7/5} or there is strong interaction between two eigenvectors in an interval of length W^{7/5}.
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Article dans une revue
Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (4), pp.DOI: 10.1214/ECP.v19-3027
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Contributeur : Florent Benaych-Georges <>
Soumis le : samedi 18 janvier 2014 - 05:40:59
Dernière modification le : jeudi 27 avril 2017 - 09:46:34
Document(s) archivé(s) le : vendredi 18 avril 2014 - 22:06:11

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  • HAL Id : hal-00863653, version 3
  • ARXIV : 1309.4922

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Florent Benaych-Georges, Sandrine Péché. Largest eigenvalues and eigenvectors of band or sparse matrices. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (4), pp.DOI: 10.1214/ECP.v19-3027. <hal-00863653v3>

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