1-2 model, dimers, and clusters

Zhongyang Li

doi:10.1214/ejp.v19-2563

Abstract

The 1-2 model is a probability measure on subgraphs of the hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We prove that for any translation-invariant Gibbs measure of the 1-2 model on the plane, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and perfect matchings on a decorated graph, we construct an explicit translation-invariant measure $P$ for 1-2 model configurations on the bi-periodic hexagonal lattice embedded into the whole plane. We prove that the behavior of infinite clusters is different for small and large local weights, which shows the existence of a phase transition.

Highlights

A 1-2 model configuration is a subgraph of the hexagonal lattice, in which the edgedegree of each vertex is either 1 or 2
We introduce a new approach to solve the 1-2 model exactly, by constructing a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and perfect matchings on a decorated graph
With the help of such a measure-preserving correspondence, we compare the behaviors of infinite homogeneous clusters for small and large local weights by analyzing the underlying perfect matchings, and prove the existence of a phase transition

Summary

Self-avoiding path and monotonicity

We prove two propositions resulting from the definition of the measure of the 1-2 model. We notice that in each 1-2 model configuration, there are two kinds of connected components: either self-avoiding paths or loops, see Figure 1. Proposition 1.1 shows how to compute the expected number of self-avoiding paths explicitly, and Proposition 1.2 shows the monotonicity of the expected number of a specific local configuration with respect to local weights. Let σn denote the number of self-avoiding paths in a random 1-2 model configuration in Hn, . For any k, k ≥ 0, 1Na≥k and 1Na≥k have positive correlation, i.e., E1Na≥k · 1Na≥k ≥ E1Na≥k · E1Na≥k , where the expectation is taken with respect to the probability measure of 1-2 model configurations on the torus Hn given periodic boundary conditions.

Nonexistence of infinite paths

Correspondence with dimers

Infinite clusters

Exponential decay of large cluster probabilities

Percolation