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

TL;DR

This paper investigates when time-evolved planar rotor models retain or lose their Gibbsian properties under stochastic dynamics, revealing conditions for Gibbsian and non-Gibbsian behavior over time.

Contribution

It provides rigorous conditions under which the evolved measures remain Gibbsian or become non-Gibbsian in planar rotor systems.

Findings

Gibbsian measures persist for small times and at long times with high or infinite temperature.

Non-Gibbsianness occurs for low-temperature initial measures during an intermediate time interval.

Results apply to systems evolving with Brownian diffusions on circles in Z^d.

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 \nu, or for long times and both high- or infinite-temperature measure and dynamics, the evolved measure \nu^t stays Gibbsian. Furthermore we show that for a low-temperature initial measure \nu, evolving under infinite-temperature dynamics thee is a time interval (t_0, t_1) such that \nu^t fails to be Gibbsian in d=2.