A Contour Method on Cayley tree

TL;DR

This paper introduces a contour method for finite range lattice models on Cayley trees, demonstrating the existence of multiple Gibbs measures corresponding to different ground states using a contour argument.

Contribution

It develops a novel contour approach tailored for Cayley trees and proves the existence of multiple Gibbs measures under specified conditions.

Findings

Existence of s distinct Gibbs measures

Contour method adapted for Cayley tree models

Validation under Peierls type condition

Abstract

We consider a finite range lattice models on Cayley tree with two basic properties: the existence of only a finite number of ground states and with Peierls type condition. We define notion of a contour for the model on the Cayley tree. By a contour argument we show the existence of $s$ different (where $s$ is the number of ground states) Gibbs measures.