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

TL;DR

This paper proves that the mixing time of the 2D stochastic Ising model with plus boundary conditions at low temperature is subexponential in the system size, significantly improving previous bounds and using innovative induction and censoring techniques.

Contribution

It establishes new subexponential bounds on the mixing time for the 2D Ising model with plus boundary conditions at low temperature, employing novel induction and censoring methods.

Findings

Mixing time is subexponential in system size for large inverse temperature.

Improves previous estimates from exponential in L^{1/2 + ε} to exponential in L^ε.

Techniques involve induction over scales and the censoring inequality.

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 $\lim_{L\to\infty}P(T_{mix}> \exp({cL^\epsilon})) =0$. In the non-random case $\tau\equiv +$ (or $\tau\equiv -$), this implies that $T_{mix}< \exp({cL^\epsilon})$. 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.