Loss without recovery of Gibbsianness during diffusion of continuous spins

Details Loss without recovery of Gibbsianness during diffusion of continuous spins Loss without recovery of Gibbsianness during diffusion of continuous spins

2004-09-23

arxiv.org/abs/math-ph/0409061v1

Physics

Mathematical Physics

Scientific paper

We consider a specific continuous-spin Gibbs distribution $\mu_{t=0}$ for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For `high temperature' initial measures we prove that the time-evoved measure $\mu_{t}$ is Gibbsian for all $t$. For `low temperature' initial measures we prove that $\mu_t$ stays Gibbsian for small enough times $t$, but loses its Gibbsian character for large enough $t$. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large $t$ in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension $d\geq 2$. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.

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