Uniqueness of Gibbs Measures for Continuous Hardcore Models

TL;DR

This paper introduces a continuous version of the hardcore model on graphs, proving uniqueness of Gibbs measures on regular trees and analyzing the model's behavior in relation to phase transitions and convex polytopes.

Contribution

It formulates a continuous hardcore model, proves the uniqueness of Gibbs measures on regular trees, and connects these results to convex polytopes and partition function approximations.

Findings

Unique Gibbs measure for each activity parameter on regular trees.

No phase transition occurs in the continuous model unlike the discrete case.

Provides asymptotic volume calculations for convex polytopes related to the model.

Abstract

We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $\lambda > 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $\lambda>0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $\lambda = 1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case, the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular, this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $\lambda = 1$.