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 $\beta$ and random boundary conditions $\tau$ 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 $\beta$ large enough we show that for any $\epsilon$ there exists $c=c(\beta ,\epsilon )$ such that the corresponding mixing time $T_{mix}$ satisfies ${\lim }_{L\to \infty }P(T_{mix}>\exp (c{L}^{\epsilon }))=0$. In the non-random case $\tau \equiv +$ (or $\tau \equiv -$), this implies that $T_{mix}<\exp (c{L}^{\epsilon })$. 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 behaviour $T_{mix}=O(L^2)$, considerably improves upon the previous known estimates of the form $T_{mix}\le \exp (c{L}^{1/2+\epsilon })$. 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.

39 pages, 8 figures; v2: typos corrected, two references added. To appear on Comm. Math. Phys

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