On the top eigenvalue of heavy-tailed random matrices

G. Biroli; J.-P. Bouchaud; M. Potters

doi:10.1209/0295-5075/78/10001

All issues

Volume 78 / No 1 (April 2007)

EPL, 78 1 (2007) 10001

Abstract

Issue		EPL Volume 78, Number 1, April 2007


Article Number		10001
Number of page(s)		5
Section		General
DOI		https://doi.org/10.1209/0295-5075/78/10001
Published online		12 March 2007

EPL, 78 (2007) 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: 10 February 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 $P(M_{ij}) \sim |M_{ij}|^{-1-\mu }$ . 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 Marčenko-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

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