On the top eigenvalue of heavy-tailed random matrices

• Published in: Europhysics letters Vol. 78; no. 1; pp. 10001 - 10001 (5)
• Main Authors: Biroli, G, Bouchaud, J.-P, Potters, M
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.04.2007; EDP Sciences
• Subjects: 02.10.Yn; 02.50.-r; 89.65.Gh
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/0295-5075/78/10001

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.