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