Asymptotic Behavior of the Expectation Value of Permanent Products

TL;DR

This paper investigates the asymptotic behavior of the expectation value of products of matrix permanents, using a model based on sums of permutation matrices, and derives limiting formulas as matrix size grows.

Contribution

It introduces a novel approach to analyze the asymptotics of permanent products through permutation matrix sums, providing new limiting expectation formulas.

Findings

Derived the limit of (1/n)ln(E(perm_m(A) perm_m'(A))) as n approaches infinity.

Established that the limit equals the sum of individual limits of perm_m(A) and perm_m'(A).

Worked with a measure based on sums of permutation matrices rather than uniform 0-1 matrices.

Abstract

We would desire to have done the calculations of this paper in the measure on nxn matrices that weights uniformly all 0-1 matrices with row and column sum equal to r, other matrices given weight zero. Instead we work with all matrices that are the sum of r independent uniformly weighted permutation matrices, with the hope that the computations we perform give the same result in this measure. We derive the result for limiting expectations lim (1/n)ln(E(perm_m(A) perm_m'(A))) =lim (1/n)ln(E(perm_m(A)))+ +lim (1/n)ln(E(perm_m'(A))) Here the limit is n to infinity, r is fixed, and m and m' are taken as each proportional to n.