Uniqueness of Infinite Homogeneous Clusters in 1-2 Model

TL;DR

This paper proves that in the 1-2 model on a hexagonal lattice, any translation-invariant Gibbs measure almost surely has a unique infinite homogeneous cluster, establishing a key property of the model's structure.

Contribution

The paper establishes the almost sure uniqueness of the infinite homogeneous cluster in the 1-2 model under translation-invariant Gibbs measures, a novel result in this area.

Findings

Almost sure uniqueness of the infinite homogeneous cluster.

Applicability to all translation-invariant Gibbs measures.

Advances understanding of the 1-2 model's phase structure.

Abstract

A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.