Four Amazing Positivities with Dimers/i-Matchings

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.)