Uniqueness of locally stable Gibbs point processes via spatial birth-death dynamics

TL;DR

This paper proves the uniqueness of infinite-volume Gibbs point processes for certain stable potentials using spatial birth-death dynamics, extending classical bounds and applying Markov process techniques.

Contribution

It introduces a new uniqueness regime for Gibbs point processes based on spatial birth-death dynamics, surpassing classical bounds by a factor of at least e.

Findings

Established a unique Gibbs measure for activity below a specific threshold.

Extended the classical Ruelle--Penrose bound for uniqueness.

Developed a Markov process approach demonstrating rapid convergence and spatial mixing.

Abstract

We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $\lambda < (e^{L} \hat{C}_{\phi})^{-1}$, where $L$ is the local stability constant and $\hat{C}_{\phi}:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|\phi(x, y)|} dy$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least $e$, where the improvements becomes larger as the negative parts of the potential become more prominent (i.e., for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.