Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Abstract

The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.(Collaborations with Benedikt Jahnel, Sascha Kissel, Utkir Rozikov) space, on the lattice, as a mean-field model, and on a regular tree. Our aim here is to provide an overview; for detailed statements and proofs we refer to the original articles.2 Gibbs on lattice, sequentially Gibbs, marked Gibbs point processes, and the Widom-Rowlinson modelWe start by recalling the notion of an infinite-volume Gibbs measure for lattice systems. For the purpose of the discussion of the Widom-Rowlinson model and all measures appearing under timeevolution defined below from it, it is sufficient to restrict to the local state-space {−1, 0, 1} for particles carrying spins plus or minus, and holes. Our site space is the lattice Z d . The space of infinite-volume configurations is Ω = {−1, 0, 1} Z d . Specifications and Gibbs measures on the latticeThe central object in Gibbsian theory on a countable site space which defines the model is a specification. This covers both cases of infinite lattices and trees. It is a candidate system for conditional probabilities of an infinite-volume Gibbs measure µ (probability measure on Ω) to be defined by DLR equations µ(γ Λ (f |·)) = µ(f ). A specification γ is by definition a family of probability kernels γ = (γ Λ ) Λ Z d , indexed by finite subvolumes Λ, where γ Λ (dω|η) is a probability measure on Ω, for each fixed configuration η. It must have the following properties. The first is the consistency which means that

Dynamical Gibbs–Non-Gibbs Transitions in Lattice Widom–Rowlinson Models with Hard-Core and Soft-Core Interactions

“…and by Lemma 4. 19 we have Ω f g(η)µ t (dη) = Ω f γ ∞ ∆ (g|η)µ t (dη) for every measurable function g and ∆ Z d .…”

“…The latter is quite unusual for a lattice model, see however the examples in mean-field [12], on a tree [2], and for a transformed measure not coming from a time-evolution in [21] Motivated by the strong anomalies which occur for the Widom-Rowlinson model in continuum, one becomes interested in the behavior of the model in other geometries: as a mean-field model, on the lattice, on a tree, on more general graphs, or in a long-range Kac-version. For a recent overview, see [19].…”