Paper detail

On the mixing time of the 2D stochastic Ising model with "plus" boundary conditions at low temperature

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 $ε$ there exists $c=c(β,ε)$ such that the corresponding mixing time $T_{mix}$ satisfies $\lim_{L\to\infty}P(T_{mix}> \exp({cL^ε})) =0$. In the non-random case $τ\equiv +$ (or $τ\equiv -$), this implies that $T_{mix}< \exp({cL^ε})$. 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 $T_{mix}\le \exp({c L^{1/2 + ε}})$. 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.