Issue
EPL
Volume 78, Number 1, April 2007
Article Number 10001
Number of page(s) 5
Section General
DOI http://dx.doi.org/10.1209/0295-5075/78/10001
Published online 12 March 2007
EPL, 78 (2007) 10001
DOI: 10.1209/0295-5075/78/10001

On the top eigenvalue of heavy-tailed random matrices

G. Biroli1, 2, J.-P. Bouchaud3, 2 and M. Potters2

1  Service de Physique Théorique, Orme des Merisiers, CEA Saclay - 91191 Gif-sur-Yvette Cedex, France
2  Science & Finance, Capital Fund Management - 6 Bd Haussmann, 75009 Paris, France
3  Service de Physique de l'État Condensé, Orme des Merisiers, CEA Saclay - 91191 Gif-sur-Yvette Cedex, France


received 16 November 2006; accepted in final form 10 February 2007; published April 2007
published online 12 March 2007

Abstract
We study the statistics of the largest eigenvalue $\lambda _{{\rm max}}$ of $N \times N$ random matrices with IID entries of variance 1/N, but with power law tails $P(M_{ij}) \sim \vert M_{ij}\vert^{-1-\mu }$. When $\mu > 4$, $\lambda _{{\rm max}}$ converges to 2 with Tracy-Widom fluctuations of order N-2/3, but with large finite N corrections. When $\mu < 4$, $\lambda _{{\rm max}}$ is of order $N^{2/\mu -1/2}$ and is governed by Fréchet statistics. The marginal case $\mu =4$ provides a new class of limiting distribution that we compute explicitly. 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.

PACS
02.10.Yn - Matrix theory.
02.50.-r - Probability theory, stochastic processes, and statistics.
89.65.Gh - Economics; econophysics, financial markets, business and management .

© Europhysics Letters Association 2007