Permanents of heavy-tailed random matrices with positive elements
arXiv:1111.3454
Abstract
We study the asymptotic behavior of permanents of random matrices with positive entries. We assume that 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.
arXiv commentv2: 26 pages, no figures, changed title, expanded the result to symmetric case