Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

Abstract

The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R d with the formal Hamiltonian defined as the volume of ∪ x∈ω 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 also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature β ≥ 0 related to the strength of the interaction. In the present paper 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 0 < z a c (β, r) < +∞ such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z < z a c (β, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z > z a c (β, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms [7,9,8]. Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, β. Old results [22,24] claim that a non-uniqueness regime occurs for z = β large enough and it is conjectured that the uniqueness should hold outside such an half line (z = β ≥ β c > 0). We solve partially this conjecture in any dimension 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 z a c (β, r) = β for β large enough, providing a straight connection between these two notions of phase transition.

An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions

“…Considering the potential to be fixed, we study the question of uniqueness depending on the parameters z, β. Heuristically, it is expected that for each β > 0, uniqueness is achieved for activities z small enough. Let us mention however that such a behaviour has actually been disproved for the specific case of the Widom-Rowlinson model with random radii with heavy tails, see [8].…”

Gibbsian Representation for Point Processes via Hyperedge Potentials

“…A set of configurations where discontinuities of finite-volume conditional probabilities of the time-evolved measure can not appear at the critical time t G , can be defined by Ω + = {ω ∈ Ω : ω has no infinite cluster C with lim inf n↑∞ |C ∩ Λ n | −1 x∈C∩Λn σ x ≤ 0}. In Table 1 we summarize the results for the asymmetric model, which is the most interesting case; see also [DH18] for recent results on the corresponding equilibrium model. In the table, when we write "no quasilocal Poisson modification" we mean that there exists no Ω ′ ⊂ Ω such that the time-evolved WRM would be concentrated on Ω ′ and there exists a quasilocal Ω ′ -Poisson modification.…”

Phase transition for the non-symmetric Continuum Potts model

“…Finally from this construction and the Proposition 4.4, the phase transition is straightforward. This type of construction is now standard and have been done in many articles, for the symmetric continuum Potts model [13] or for other types of interaction [5,6,7,9].…”

“…This type of construction is classical. It was done for instance for the symmetric Potts model in [13], for the quermass-interaction model in [5], and for many other cases [6,7,9]. The first step is to construct a good candidate.…”

“…In the 1990' Chayes, Chayes & Kotecký [3] and Georgii & Häggström [13] generalized for continuum models the idea of the Fortuin-Kasteleyn representation [10] introduced for the lattice Ising and Potts models, and proved phase transition results respectively for the symmetric Widom-Rowlinson model and for the continuum Potts model with background interaction. This idea was then used in a variety of articles, for instance to prove phase transition for the symmetric Widom-Rowlinson model with unbounded radii [9,17]. The idea of the Fortuin-Kasteleyn representation is generalized to the non-symmetric case in the present article.…”