On the top eigenvalue of heavy-tailed random matrices

Abstract

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 Fréchet 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.

4 pages, 2 figures

References in corpus (3)

• Exact scaling functions for one-dimensional stationary KPZ growth
• The largest eigenvalue of rank one deformation of large Wigner matrices
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

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