On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

Abstract

We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to −). For β large enough we show that for any \({\varepsilon >0 }\) there exists \({c=c(\beta,\varepsilon)}\) such that the corresponding mixing time T mix satisfies \({{\rm lim}_{L\to\infty}\,{\bf P}\left(T_{\rm mix}\ge {\rm exp}({cL^\varepsilon})\right) =0}\). In the non-random case τ ≡ + (or τ ≡ −), this implies that \({T_{\rm mix}\le {\rm exp}({cL^\varepsilon})}\). The same bound holds when the boundary conditions are all + on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior T mix = O(L 2), considerably improves upon the previous known estimates of the form \({T_{\rm mix}\le {\rm exp}({c L^{\frac 12 + \varepsilon}})}\). The techniques are based on induction over length scales, combined with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.