The Annals of Applied Probability

The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

Benedikt Jahnel and Christof Külske

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Abstract

We consider the continuum Widom–Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs–non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices.

We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-almost-sure quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time $t_{G}>0$, the model is a.s. quasilocal. For the color-symmetric model, there is no reentrance. On the constructive side, for all $t>t_{G}$, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3845-3892.

Dates
Received: September 2016
Revised: March 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328715

Digital Object Identifier
doi:10.1214/17-AAP1298

Mathematical Reviews number (MathSciNet)
MR3737939

Zentralblatt MATH identifier
06848280

Subjects
Primary: 82C21: Dynamic continuum models (systems of particles, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Gibbsianness non-Gibbsianness point processes Widom–Rowlinson model spin-flip dynamics quasilocality non-almost-sure quasilocality $\tau$-topology

Citation

Jahnel, Benedikt; Külske, Christof. The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27 (2017), no. 6, 3845--3892. doi:10.1214/17-AAP1298. https://projecteuclid.org/euclid.aoap/1513328715


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