On the top eigenvalue of heavy-tailed random matrices

We study the statistics of the largest eigenvalue lambda(max) of N x N random matrices with IID entries of variance 1/ N, but with power law tails P(M(ij)) similar to vertical bar M(ij)vertical bar(-1-mu). When mu > 4, lambda(max) converges to 2 with Tracy-Widom fluctuations of order N(-2/3), but with large finite N corrections. When mu < 4, lambda(max) is of order N(2/mu-1/2) and is governed by Frechet 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. Copyright (c) EPLA, 2007.