Approximate Lifshitz law for the mixing time of the zero-temperature stochastic Ising model with + boundary conditions in any dimension
ABSTRACT
It has been noticed that below te critical temperature the mixing properties of the Stochastic Ising model are strongly dependent on boundary condition. Indeed if one considers Heat-Bath Dynamics for Ising in the cube of side L, the mixing time is exponential in L whereas it is believed that for all + boundary condition, it behaves like L^2 (conjecture called "Lifshitz law"). What we present here is a new step toward the verification of the conjecture, showing that in all dimension, the mixing time with + boundary condition is O(L^2 \log L^c) for soma appropriate c in any dimention. This generalizes a recent result by Caputo, Martinelli, Simenhaus and F.L.Toninelli (who proved it in two and three dimension). We will present the key results obtained by Caputo et al. for the three dimensional model and explain how they can be used to obtained the result for dimension larger than three.