Extreme eigenvalues of sparse, heavy tailed random matrices

We study the statistics of the largest eigenvalues of p×p sample covariance matrices Σp,n=Mp,nMp,n∗ when the entries of the p×n matrix Mp,n are sparse and have a distribution with tail t−α, α>0. On average the number of nonzero entries of Mp,n is of order nμ+1, 0≤μ≤1. We prove that in the large n limit, the largest eigenvalues are Poissonian if α<2(1+μ−1) and converge to a constant in the case α>2(1+μ−1). We also extend the results of Benaych-Georges and Péché (2014) in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.