Fixed points of an infinite dimensional operator related to Gibbs measures

TL;DR

This paper analyzes fixed points of a complex infinite-dimensional operator linked to a hard core model on a Cayley tree, establishing conditions for uniqueness and multiplicity of fixed points and their relation to Gibbs measures.

Contribution

It provides a sufficient condition for the uniqueness of fixed points and demonstrates the potential for up to five fixed points when this condition fails.

Findings

Unique fixed point under certain parameter conditions

Possibility of up to five fixed points otherwise

Each fixed point corresponds to a Gibbs measure

Abstract

We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters $\lambda_{i}>0$, $a_{ij}\in \{0,1\}$, $ i,j \in \mathbb{N}$. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs measure for the given HC-model.