Potential-weighted connective constants and uniqueness of Gibbs measures

TL;DR

This paper introduces a potential-weighted connective constant to analyze Gibbs point processes, providing improved bounds for uniqueness of Gibbs measures in various spaces, and demonstrating tightness of these bounds in tree-like geometries.

Contribution

The paper defines a new potential-weighted connective constant and uses it to establish sharper bounds for Gibbs measure uniqueness across different metric spaces.

Findings

Improved bounds for Gibbs uniqueness in Euclidean and metric measure spaces.

Construction of a tree-branching collection of densities capturing potential and geometry.

Identification of tightness of bounds in tree-like geometries.

Abstract

We define a potential-weighted connective constant that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on $\mathbb R^d$ and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.