2D Ising and dimer models

Abstract

Since the solution of the 2-dimensional (planar) Ising problem was achieved by Onsager, the physicists have been trying to reproduce his solution by more understandable methods. In the fifties and in the beginning of sixties two discrete methods appeared: the Pfaffian method of Kasteleyn and independently Fisher, Temperley, and the path method of Kac, Ward, Potts, Feynman and Sherman. Both methods start by reducing the Ising partition function Z(G, β) to the generating function ε(G, x) of even subsets of edges. This is accomplished by van der Waerden's theorem (Theorem 6.3.1). The Pfaffian method seems to be better known to discrete mathematicians. It further reduces ε(G, x) to the generating function P(G′, x) of the perfect matchings (dimer arrangements) of a graph G′ obtained from G by a local operation at each vertex (see Section 6.2). It is important that these operations are locally planar, i.e., G′ may be embedded on the same surface as G.