2 research outputs found
Spectral Density of Sparse Sample Covariance Matrices
Applying the replica method of statistical mechanics, we evaluate the
eigenvalue density of the large random matrix (sample covariance matrix) of the
form , where is an real sparse random matrix.
The difference from a dense random matrix is the most significant in the tail
region of the spectrum. We compare the results of several approximation
schemes, focusing on the behavior in the tail region.Comment: 22 pages, 4 figures, minor corrections mad
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 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 Fr\'echet 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.Comment: 4 pages, 2 figure