Permanents of heavy-tailed random matrices with positive elements

TL;DR

This paper investigates the asymptotic behavior of permanents of large random matrices with positive, heavy-tailed entries, revealing phase transitions and uniform limits across submatrices.

Contribution

It establishes a strong law of large numbers for the logarithm of the permanent in heavy-tailed regimes and characterizes the limit in terms of tail decay exponents.

Findings

Proves a strong law of large numbers for log permanents with heavy-tailed entries.

Calculates the limit of log perm divided by n log n based on tail decay exponent.

Shows uniform convergence over all submatrices of linear size in the finite mean regime.

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.