Liquid-Gas phase transition for Gibbs point process with Quermass interaction

TL;DR

This paper demonstrates the existence of a liquid-gas phase transition in a continuous Gibbs point process with Quermass interaction, showing non-uniqueness of Gibbs measures and pressure non-differentiability at critical points.

Contribution

It extends phase transition theory to continuous systems with complex geometric interactions using an adapted Pirogov-Sina"i-Zahradnik approach.

Findings

Non-uniqueness of Gibbs measures at low temperature and specific activity levels.

Non-differentiability of pressure at critical points.

Existence of a liquid-gas phase transition in the model.

Abstract

We prove the existence of a liquid-gas phase transition for continuous Gibbs point process in $\mathbb{R}^d$ with Quermass interaction. The Hamiltonian we consider is a linear combination of the volume $\mathcal{V}$, the surface measure $\mathcal{S}$ and the Euler-Poincar\'e characteristic $\chi$ of a halo of particles (i.e. an union of balls centred at the positions of particles). We show the non-uniqueness of infinite volume Gibbs measures for special values of activity and temperature, provided that the temperature is low enough. Moreover we show the non-differentiability of the pressure at these critical points. Our main tool is an adaptation of the Pirogov-Sina\"i-Zahradnik theory for continuous systems with interaction exhibiting a saturation property.