A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0-1 Matrices

TL;DR

This paper investigates the expectation value of the permanent of 0-1 matrices across different ensembles, revealing a mysterious cluster expansion with remarkable properties and conjecturing its validity for all terms.

Contribution

It introduces a novel cluster expansion relating expectations of permanents in two matrix ensembles and extends previous results using recent graph matching theories.

Findings

E(perm(A)) = EB(perm(A)) * exp{sum Ti} relation established

Ti terms match for i < 21 across ensembles, extending previous results

Conjecture that properties of Ti hold for all i

Abstract

We consider two ensembles of nxn matrices. The first is the set of all nxn matrices with entries zeroes and ones such that all column sums and all row sums equal r, uniformly weighted. The second is the set of nxn matrices with zero and one entries where the probability that any given entry is one is r/n, the probabilities of the set of individual entries being i.i.d.'s. Calling the two expectation values E and EB respectively, we develop a formal relation E(perm(A)) = EB(perm (A)) x exp{sum Ti}. We also use a well-known approximating ensemble to E, E1. We prove using E or E1 one obtains the same value of Ti for i < 21. (THE PUBLISHED VERSION OF THIS PAPER ONLY OBTAINS RESULTS FOR i < 8. We go beyond the results of the published version by taking much more advantage of recent work of Pernici and of Wanless on i-matchings on regular bipartite graphs.)These terms Ti, i < 21, have amazing properties. We conjecture that these properties hold also for all i.