Permanents through probability distributions

TL;DR

This paper presents a probabilistic approach to expressing the matrix permanent as an expectation over random variables, extending known theorems and linking the permanent to spin system partition functions.

Contribution

It introduces a novel probabilistic representation of the permanent, unifies existing theorems, and derives a new relation involving hyperbolic functions and spin systems.

Findings

Permanent expressed as expectation over zero-mean, unit-variance variables

Extension of Glynn's and MacMahon theorems via probabilistic interpretation

New relation between permanent and spin system partition function

Abstract

We show that the permanent of a matrix can be written as the expectation value of a function of random variables each with zero mean and unit variance. This result is used to show that Glynn's theorem and a simplified MacMahon theorem extend from a common probabilistic interpretation of the permanent. Combining the methods in these two proofs, we prove a new result that relates the permanent of a matrix to the expectation value of a product of hyperbolic trigonometric functions, or, equivalently, the partition function of a spin system. We conclude by discussing how the main theorem can be generalized and how the techniques used to prove it can be applied to more general problems in combinatorics.