Gibbsianness versus Non-Gibbsianness of time-evolved planar rotor models

2007/11/22 by Aernout C. D. van Enter, A. C. D. van Enter, van Enter, A. C. D. +3 · 2 citations
Mathematics · Physics and Astronomy · #82c20 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:82c20

paper · pdf · doi:10.48550/arxiv.0711.3621

latexpdf, with 2 pdf figures

arxiv created 2007/11/22 · openalex publication_date 2007/11/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Gibbsian character of time-evolved planar rotor systems on Z^d, d at least 2, in the transient regime, evolving with stochastic dynamics and starting with an initial Gibbs measure. We model the system by interacting Brownian diffusions, moving on circles. We prove that for small times and arbitrary initial Gibbs measures ν, or for long times and both high- or infinite-temperature measure and dynamics, the evolved measure ν^t stays Gibbsian. Furthermore we show that for a low-temperature initial measure ν, evolving under infinite-temperature dynamics thee is a time interval (t₀, t₁) such that ν^t fails to be Gibbsian in d=2.

Cited by

• Gibbsian and non-Gibbsian states at Eurandom
• Gibbs-non-Gibbs properties for n-vector lattice and mean-field models

Related

• Gibbs-non-Gibbs properties for n-vector lattice and mean-field models
• Gibbs-non-Gibbs properties for evolving Ising models on trees
• Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models
• Loss without recovery of Gibbsianness during diffusion of continuous spins
• Stationary properties of a Brownian gyrator with non-Markovian baths