Publication : t06/216
On the top eigenvalue of heavy-tailed random matrices
Biroli G. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)Bouchaud J.-P. (CEA, IRAMIS, SPEC (Service de Physique de lEtat Condensé), F-91191 Gif-sur-Yvette, FRANCE)
Potters M. (Science & Finance, Capital Fund Management 6 Bld Haussmann, F-75009 Paris, France)
Abstract:Année de publication : 2007
We study the statistics of the largest eigenvalue lambda_max of N x N random matrices with unit variance, but power-law distributed entries, P(M_{ij})~ |M_{ij}|^{-1-mu}. When mu > 4, lambda_max converges to 2 with Tracy-Widom fluctuations of order N^{-2/3}. When mu < 4, lambda_max is of order N^{2/mu-1/2} and is governed by Fr\'echet statistics. The marginal case mu=4 provides a new class of limiting distribution that we compute explicitely. We extend these results to sample covariance matrices, and show that extreme events may cause the largest eigenvalue to significantly exceed the Marcenko-Pastur edge. Connections with Directed Polymers are briefly discussed.
Revue : Europhys. Lett. 78 10001 (2007)
DOI : 10.1209/0295-5075/78/10001
Preprint : arXiv:cond-mat/0609070
Lien : http://www.iop.org/EJ/abstract/-search=29582366.4/0295-5075/78/1/10001
Numéro Exterieur : SPEC-S06/134
Langue : Anglais
Fichier(s) à télécharger :
publi.pdf
