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

We study the Gibbsian character of time-evolved planar rotor systems (that is, systems which have two-component, classical XYXY, spins) on ZdZd, d≥2d≥2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure νν. We model the system with interacting Brownian diffusions X=(Xi(t))t≥0,i∈ZdX=(Xi(t))t≥0,i∈Zd moving on circles. We prove that for small times tt and arbitrary initial Gibbs measures νν, or for long times and both high- or infinite-temperature initial measure and dynamics, the evolved measure νtνt stays Gibbsian. Furthermore, we show that for a low-temperature initial measure νν evolving under infinite-temperature dynamics there is a time interval (t0,t1)(t0,t1) such that νtνt fails to be Gibbsian for d≥2d≥2.