Continuum percolation for Gibbsian point processes with attractive interactions

TL;DR

This paper investigates continuum percolation in Gibbs measures with attractive interactions, providing bounds on percolation thresholds and a variational formula linking cluster size distributions to Gibbs principles.

Contribution

It introduces bounds on percolation thresholds based on density and derives a variational formula connecting cluster distributions with Gibbs variational principles.

Findings

Bounds on percolation thresholds in terms of density

A variational formula for large deviations of cluster sizes

Link between finite and infinite volume Gibbs measures

Abstract

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large $\beta$). The main results are bounds on percolation thresholds $\rho_\pm(\beta)$ in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures.