Potential-weighted connective constants and uniqueness of Gibbs measures

2021/09/02 by Marcus Michelen, Michelen, Marcus, Will Perkins +1 · 2 citations
Mathematics · #Point processes and geometric inequalities #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2109.01094

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.

Cited by

• A phase transition for the hard sphere model on the hyperbolic plane
• Simple analyticity criteria for repulsive multi-body potentials
• Upper bounds for the connective constant of weighted self-avoiding walks

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