Permanents of heavy-tailed random matrices with positive elements

Abstract

We study the asymptotic behavior of permanents of $n \times n$ random matrices $A$ with positive entries. We assume that $A$ has either i.i.d. entries or is a symmetric matrix with the i.i.d. upper triangle. Under the assumption that elements have power law decaying tails, we prove a strong law of large numbers for $\log \perm A$. We calculate the values of the limit $\lim_{n \to \infty}\frac{\log \perm A}{n \log n}$ in terms of the exponent of the power law distribution decay, and observe a first order phase transition in the limit as the mean becomes infinite. The methods extend to a wide class of rectangular matrices. It is also shown that, in finite mean regime, the limiting behavior holds uniformly over all submatrices of linear size.