Abstract
We study the statistics of the largest eigenvalue λmax of N × N random matrices with IID entries of variance 1/N, but with power law tails P(Mij) ∼ |Mij|−1−μ. When μ > 4, λmax converges to 2 with Tracy-Widom fluctuations of order N−2/3, but with large finite N corrections. When μ < 4, λmax is of order N2/μ−1/2 and is governed by Fréchet statistics. The marginal case μ = 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.