Gibbs-non-Gibbs properties for n-vector lattice and mean-field models

We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.Comment: 31 pages, 6 figure

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

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.Comment: latexpdf, with 2 pdf figure

Loss and Recovery of Gibbsianness for XY models in external fields

We consider planar rotors (XY spins) in $\mathbb{Z}^d$, starting from an initial Gibbs measure and evolving with infinite-temperature stochastic (diffusive) dynamics. At intermediate times, if the system starts at low temperature, Gibbsianness can be lost. Due to the influence of the external initial field, Gibbsianness can be recovered after large finite times. We prove some results supporting this picture.Comment: 14 pages, 2 figure

Discrete approximations to vector spin models

We strengthen a result of two of us on the existence of effective interactions for discretised continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretising continuous-spin models, and show that, except for very low temperatures, they behave similarly in two dimensions. We also discuss some possibilities in higher dimensions.Comment: 12 page

Chaotic temperature dependence at zero temperature

We present a class of examples of nearest-neighbour, boubded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension