Perfect matchings of fisher graphs of cubic graphs

摘要

Fisher's solution of the Ising model on a planar graph G relies on converting it to a dimer model on a related planar graph G~Δ, which in turn can be solved using Pfaffians. In this paper we show that for cubic graphs G the number of perfect matchings of the graph G~Δ equals 2~(|E(G)|-|V (G)|+1). Our proof is based on two combinatorial ideas from statistical physics, namely the high-temperature expansion of the Ising model, and the Fisher construction that maps the Ising model to a dimer model.We give several applications, including two concrete three-dimensional crystal structures whose entropy can be computed exactly using our result.