Permanents of heavy-tailed random matrices with positive elements

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.