Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

Abstract:

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:

(1) For all ν and μ, νS(t) is Gibbs for small t.

(2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.

(3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.

(4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t.

The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.