Four Amazing Positivities with Dimers/i-Matchings

TL;DR

This paper surveys recent results on dimers and i-matchings on bipartite lattices and graphs, highlighting proven findings and open conjectures to encourage further research in the field.

Contribution

It clearly distinguishes proven results from conjectures and identifies four research directions, including positivity of virial coefficients and graph positivity for small bipartite graphs.

Findings

First 20 virial coefficients are positive for all hyper-rectangular lattices.

All regular bipartite graphs with fewer than 14 vertices satisfy graph positivity.

Abstract

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show people: here is a field which is ripe for further interesting research. We separate four classes of endeavor, of which we here extract two items to whet one's appetite. Primo,for hyper-rectangular lattices of every dimension the first 20 virial coefficients are positive. (One has no understanding of this yet!) Secondo, all regular bipartite graphs with less than $14$ vertices satisfy graph positivity, defined below. (Here there is some understanding.)