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

David Dereudre, Christopher Renaud-Chan

TL;DR

The paper addresses liquid-gas type phase transitions for a continuum Gibbs point process with Quermass interaction by translating the PSZ contour method to a saturation-enabled continuum setting. The authors construct a rigorous polymer/cluster expansion around coarse-grained states, establishing a Peierls-type energy bound and proving the existence of two distinct infinite-volume Gibbs measures at a critical activity $z_β^c$ when $\beta$ is large. The main contributions are the adaptation of PSZ theory to continuous systems with halo-based Minkowski functionals, the demonstration of non-differentiability of the pressure at the transition, and the precise control of truncated weights leading to a first-order transition without invoking spins. This advances understanding of phase structure in geometric continuum models and provides a robust framework for other saturated-interaction systems.

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-Poincaré characteristic $χ$ 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-Sinaï-Zahradnik theory for continuous systems with interaction exhibiting a saturation property.

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

TL;DR

The paper addresses liquid-gas type phase transitions for a continuum Gibbs point process with Quermass interaction by translating the PSZ contour method to a saturation-enabled continuum setting. The authors construct a rigorous polymer/cluster expansion around coarse-grained states, establishing a Peierls-type energy bound and proving the existence of two distinct infinite-volume Gibbs measures at a critical activity when is large. The main contributions are the adaptation of PSZ theory to continuous systems with halo-based Minkowski functionals, the demonstration of non-differentiability of the pressure at the transition, and the precise control of truncated weights leading to a first-order transition without invoking spins. This advances understanding of phase structure in geometric continuum models and provides a robust framework for other saturated-interaction systems.

Abstract

We prove the existence of a liquid-gas phase transition for continuous Gibbs point process in with Quermass interaction. The Hamiltonian we consider is a linear combination of the volume , the surface measure and the Euler-Poincaré characteristic 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-Sinaï-Zahradnik theory for continuous systems with interaction exhibiting a saturation property.
Paper Structure (15 sections, 16 theorems, 186 equations, 1 figure)

This paper contains 15 sections, 16 theorems, 186 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quermass interaction model
  3. State spaces and notations
  4. Interaction
  5. Gibbs measures
  6. Results
  7. Proof
  8. Coarse Graining decomposition of the energy
  9. Contours
  10. Partition function and boundary condition
  11. Energy and Peierls condition
  12. Truncated weights and pressure
  13. Proof of Proposition \ref{['prop_first_order_transition']}
  14. Annex A : Cluster Expansion
  15. Annex B : Proof of Proposition \ref{['prop_tau_stability_truncated_weights']}

Key Result

Theorem 1

Let $\theta_1,\theta_2$ be two parameters such that $\theta_1 > -\theta_1^*$ and $0 \leq \theta_2 < \theta_2^*(\theta_1)$ (recall that $\theta_2=0$ if $d\ge 3$). Then there exists $\beta_c(\theta_1, \theta_2) > 0$ such that for all $\beta > \beta_c(\theta_1, \theta_2)$, there exists $z_\beta^c > 0$

Figures (1)

  • Figure 1: The contour corresponds to the grey areas, while the blue and red squares represent the tiles at the boundary of the contour where the spins are $\#$ and $1-\#$, respectively. In Figure \ref{['fig_a']}, the contour $\Gamma =\{\gamma_1, \gamma_2\}$ is achievable by some configuration $\omega$ because the label of $A$ matches for both $\gamma_1$ and $\gamma_2$. In contrast, in Figure \ref{['fig_b']}, the contour $\Gamma = \{\gamma_1, \gamma_2\}$ is not globally achievable by any configuration. In this case, $\Gamma \in \mathcal{C}^\#(\Lambda)$, where the types of $\gamma_1$ and $\gamma_2$ are the same, but the labels of $A$ are mismatched.

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Proposition 1
  • Remark
  • Lemma 2
  • Definition 4
  • Definition 5
  • Lemma 3
  • ...and 26 more