Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges; Sandrine Péché

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1385-1403
  • ISSN: 0246-0203

Abstract

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We consider some random band matrices with band-width N μ whose entries are independent random variables with distribution tail in x - α . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α l t ; 2 ( 1 + μ - 1 ) , the largest eigenvalues have order N ( 1 + μ ) / α , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov in (Electron. Comm. Probab. 9 (2004) 82–91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles(2006) 351–364) when α l t ; 2 and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist.45(2005) 589–610) when α l t ; 4 ). On the other hand, when α g t ; 2 ( 1 + μ - 1 ) , the largest eigenvalues have order N μ / 2 and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their N coordinates.

How to cite

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Benaych-Georges, Florent, and Péché, Sandrine. "Localization and delocalization for heavy tailed band matrices." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1385-1403. <http://eudml.org/doc/272016>.

@article{Benaych2014,
abstract = {We consider some random band matrices with band-width $N^\{\mu \}$ whose entries are independent random variables with distribution tail in $x^\{-\alpha \}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha &lt;2(1+\mu ^\{-1\})$, the largest eigenvalues have order $N^\{(1+\mu )/\alpha \}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov in (Electron. Comm. Probab. 9 (2004) 82–91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles(2006) 351–364) when $\alpha &lt;2$ and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist.45(2005) 589–610) when $\alpha &lt;4$). On the other hand, when $\alpha &gt;2(1+\mu ^\{-1\})$, the largest eigenvalues have order $N^\{\mu /2\}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.},
author = {Benaych-Georges, Florent, Péché, Sandrine},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random matrices; band matrices; heavy tailed random variables; eigenvalue; eigenvector; phase transition; Poisson process},
language = {eng},
number = {4},
pages = {1385-1403},
publisher = {Gauthier-Villars},
title = {Localization and delocalization for heavy tailed band matrices},
url = {http://eudml.org/doc/272016},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Benaych-Georges, Florent
AU - Péché, Sandrine
TI - Localization and delocalization for heavy tailed band matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1385
EP - 1403
AB - We consider some random band matrices with band-width $N^{\mu }$ whose entries are independent random variables with distribution tail in $x^{-\alpha }$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha &lt;2(1+\mu ^{-1})$, the largest eigenvalues have order $N^{(1+\mu )/\alpha }$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov in (Electron. Comm. Probab. 9 (2004) 82–91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles(2006) 351–364) when $\alpha &lt;2$ and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist.45(2005) 589–610) when $\alpha &lt;4$). On the other hand, when $\alpha &gt;2(1+\mu ^{-1})$, the largest eigenvalues have order $N^{\mu /2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.
LA - eng
KW - random matrices; band matrices; heavy tailed random variables; eigenvalue; eigenvector; phase transition; Poisson process
UR - http://eudml.org/doc/272016
ER -

References

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