Potential-Weighted Connective Constants and Uniqueness of Gibbs Measures

Abstract

We define a new ‘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.