Extreme eigenvalues of sparse, heavy tailed random matrices

We study the statistics of the largest eigenvalues of $p times p$ sample covariance matrices $Sigma_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-alpha}$, $alpha>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{mu+1}$, $0 leq mu leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $alpha<2(1+mu^{{-1}})$ and converge to a constant in the case $alpha>2(1+mu^{{-1}})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.