Continuum percolation for Gibbs point processes

TL;DR

This paper investigates the percolation behavior of Boolean models generated by Gibbs point processes, establishing conditions for the existence or absence of percolation based on activity levels.

Contribution

It provides new theoretical results on percolation thresholds for Gibbs point processes, extending understanding of phase transitions in spatial stochastic models.

Findings

Percolation occurs almost surely above a critical activity for a broad class of Gibbs processes.

Locally stable Gibbs processes do not percolate at low activity levels.

The paper identifies conditions under which percolation is guaranteed or absent.

Abstract

We consider percolation properties of the Boolean model generated by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical activity, such that percolation occurs a.s. above criticality. For locally stable Gibbs point processes we show a converse result, i.e. they do not percolate a.s. at low activity.