[Paper Review] Sharp phase transition for the continuum Widom-Rowlinson model

David Dereudre, Pierre Houdebert|arXiv (Cornell University)|Jul 13, 2018

Stochastic processes and statistical mechanics31 references5 citations

TL;DR

This paper establishes a sharp phase transition in the continuum Widom-Rowlinson model by proving exponential decay of connectivity in the subcritical phase and a linear lower bound on infinite cluster connectivity in the supercritical phase, using a novel application of randomized tree algorithms and an OSSS inequality. It further shows that the non-uniqueness of Gibbs states (liquid-gas transition) occurs precisely at the percolation threshold $ z = \beta $ for large $ \beta $, resolving a key conjecture in the field.

ABSTRACT

The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in $\mathbb{R}^d$ with the formal Hamiltonian $H(ω)= ext{Volume}(\cup_{x\inω} B_1(x))$, where $ω$ is a locally finite configuration of points and $B_1(x)$ denotes the unit closed ball centred at $x$. The model is tuned by two parameters: the activity $z>0$ and the inverse temperature $β\ge 0$. We investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than $2r>0$, we show that for any $β>0$, there exists $00$). We solve partially this conjecture by showing that for $β$ large enough the non-uniqueness holds if and only if $z=β$. We show also that this critical value $z=β$ corresponds to the percolation threshold $ ilde{z}^a(β, r)=β$ for $β$ large enough, providing a straight connection between these two notions of phase transition.

Motivation & Objective

• To establish a sharp phase transition in the continuum Widom-Rowlinson model using percolation theory.
• To resolve a long-standing conjecture on the uniqueness/non-uniqueness of Gibbs states by showing non-uniqueness holds if and only if $ z = \beta $ for large $ \beta $.
• To connect the percolation threshold $ \widetilde{z}_{c}^{a}(\beta,r) $ with the liquid-gas phase transition at $ z = \beta $.
• To develop and apply a randomized tree algorithm framework to interacting continuum particle systems, proving a new OSSS inequality for the model.

Proposed method

• The authors use a randomized tree algorithm to analyze connectivity in the continuum Widom-Rowlinson model, constructing a coupling between Gibbs measures with different boundary conditions.
• They prove a new OSSS-type inequality for the Widom-Rowlinson model, enabling control over the influence of individual points on global connectivity.
• The method relies on a recursive exploration of point configurations in cubes, using a deterministic rule to select the next cube based on proximity to already explored regions.
• The construction ensures almost sure termination by leveraging the positive lower bound on the probability of empty configurations, which leads to a geometrically distributed number of steps.
• The proof of exponential decay and linear lower bounds on connectivity uses Palm theory to relate point process properties to cluster size behavior.
• The analysis of the liquid-gas transition involves studying the derivative of the partition function and applying a standard derivative theorem to derive a key identity for the derivative of the Gibbs measure.

Experimental results

Research questions

• RQ1 Does the continuum Widom-Rowlinson model exhibit a sharp phase transition in terms of percolation connectivity?
• RQ2 Is the non-uniqueness of Gibbs states in the model equivalent to the percolation threshold, particularly at $ z = \beta $ for large $ \beta $?
• RQ3 Can the randomized tree algorithm framework be adapted to interacting continuum particle systems to prove sharp phase transitions?
• RQ4 What is the relationship between the percolation threshold $ \widetilde{z}_{c}^{a}(\beta,r) $ and the liquid-gas transition point?

Key findings

• For any $ \beta > 0 $, there exists a critical activity $ \widetilde{z}_{c}^{a}(\beta,r) \in (0, \infty) $ such that exponential decay of connectivity occurs when $ z < \widetilde{z}_{c}^{a}(\beta,r) $.
• In the supercritical phase $ z > \widetilde{z}_{c}^{a}(\beta,r) $, the probability of connection to infinity is bounded below by a linear function of $ z - \widetilde{z}_{c}^{a}(\beta,r) $, implying a positive density of the infinite cluster.
• The function $ \beta \mapsto \widetilde{z}_{c}^{a}(\beta,r) $ is non-decreasing and Lipschitz continuous.
• For large $ \beta $, non-uniqueness of Gibbs states occurs if and only if $ z = \beta $, confirming a key conjecture in the literature.
• The critical value $ z = \beta $ for non-uniqueness coincides exactly with the percolation threshold $ \widetilde{z}_{c}^{a}(\beta,r) $ for large $ \beta $, establishing a direct link between the two phase transitions.