The 1-2 model: dimers, polygons, the Ising model, and phase transition

Geoffrey R. Grimmett, Zhongyang Li

Published 2015-07-15Version 1

The 1-2 model on the hexagonal lattice is a process of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was proposed in a study by Schwartz and Bruck of constrained coding systems, and is strongly connected to the dimer model on a decorated graph, and to an enhanced Ising model and an associated polygon model on the graph derived from the hexagonal lattice by adding a further vertex in the middle of each edge. The current paper is a short review of rigorous results and open problems for the 1-2 model. The general 1-2 model possesses three parameters $a$, $b$, $c$. The fundamental technique is to represent probabilities of interest as ratios of counts of dimer coverings of certain associated graphs, and to apply the Pfaffian method of Kasteleyn, Temperley, and Fisher. This approach yields certain exact representations, as well as results in the infinite-volume limit. Of especial interest is the existence (or not) of phase transitions. It turns out that all clusters of the infinite-volume limit are almost surely finite. On the other hand, the existence (with strictly positive probability) of infinite 'homogenous' clusters, containing vertices of given type, depends on the values of the parameters. A further type of phase transition emerges in the study of the two-edge correlation function, and in this case the critical surface may be found explicitly. For instance, when $a \ge b \ge c > 0$, the surface given by $\sqrt a = \sqrt b + \sqrt c$ is critical.