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.