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Phase transitions for the Widom--Rowlinson model in random environments

Benedikt Jahnel, Daniel Kamecke

TL;DR

This work extends the phase-transition theory for the continuum two-colored Widom--Rowlinson model to inhomogeneous and random environments. By leveraging a random-cluster representation and FKG domination, the authors relate non-uniqueness of infinite-volume Gibbs measures to a-percolation of the underlying point process, proving almost-sure non-uniqueness for Cox processes with stationary intensity distributions in suitable regimes. The results cover both fixed inhomogeneous environments and random environments, yielding explicit percolation-based criteria (via a dimension-dependent constant $\tau$) that guarantee multiple Gibbs measures and hence a phase transition. The approach also encompasses stabilizing, asymptotically essentially connected random environments, ensuring percolation and almost-sure non-uniqueness for a broad class of Cox point processes, with concrete examples including absolutely continuous and singular environments such as Poisson–Voronoi tessellations and Poisson–Manhattan grids. The findings connect continuum Gibbs theory with percolation and random-field analogies, providing robust tools for understanding phase behavior in spatially heterogeneous media.

Abstract

We establish non-uniqueness regimes for the infinite-volume two-colored Widom--Rowlinson model based on inhomogeneous Poisson point processes with locally finite intensity measures featuring percolation. As an application, we provide almost-sure phase-transition results for the Widom--Rowlinson model based on translation-invariant and ergodic Cox point processes with stabilizing and non-stabilizing directing measures.

Phase transitions for the Widom--Rowlinson model in random environments

TL;DR

This work extends the phase-transition theory for the continuum two-colored Widom--Rowlinson model to inhomogeneous and random environments. By leveraging a random-cluster representation and FKG domination, the authors relate non-uniqueness of infinite-volume Gibbs measures to a-percolation of the underlying point process, proving almost-sure non-uniqueness for Cox processes with stationary intensity distributions in suitable regimes. The results cover both fixed inhomogeneous environments and random environments, yielding explicit percolation-based criteria (via a dimension-dependent constant ) that guarantee multiple Gibbs measures and hence a phase transition. The approach also encompasses stabilizing, asymptotically essentially connected random environments, ensuring percolation and almost-sure non-uniqueness for a broad class of Cox point processes, with concrete examples including absolutely continuous and singular environments such as Poisson–Voronoi tessellations and Poisson–Manhattan grids. The findings connect continuum Gibbs theory with percolation and random-field analogies, providing robust tools for understanding phase behavior in spatially heterogeneous media.

Abstract

We establish non-uniqueness regimes for the infinite-volume two-colored Widom--Rowlinson model based on inhomogeneous Poisson point processes with locally finite intensity measures featuring percolation. As an application, we provide almost-sure phase-transition results for the Widom--Rowlinson model based on translation-invariant and ergodic Cox point processes with stabilizing and non-stabilizing directing measures.
Paper Structure (11 sections, 9 theorems, 31 equations, 2 figures)

This paper contains 11 sections, 9 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Two-colored Widom--Rowlinson models in inhomogeneous environments
  4. Main results
  5. Examples
  6. Proofs
  7. Existence of infinite-volume Widom--Rowlinson models
  8. Random cluster model
  9. FKG domination
  10. Non-uniqueness of the Widom--Rowlinson model
  11. Widom--Rowlinson models in random environments

Key Result

Theorem 2.1

Let $a>0$ and $\sigma$ a locally finite measure on $\mathbb{R}^d$. There is a $\tau>0$, only dependent on the dimension $d\ge 1$, such that

Figures (2)

  • Figure 1: Examples of random environments.
  • Figure 2: Realization of the two-colored Widom--Rowlinson model based on Cox point process with intensity measure given by a Poisson--Voronoi tessellation.

Theorems & Definitions (21)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Example 3.1: Non-random measures
  • Example 3.2: Absolutely continuous random environments
  • Example 3.3: Singular random environments
  • Corollary 3.4
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 11 more