21-29 Paul Federbush

Four Amazing Positivities with Dimers/i-matchings (11K, LaTeX) May 24, 21

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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 dimension d, 0 < d < 11, 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.)

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