Rapid phase ordering of Ising dynamics on $\mathbb Z^2$

TL;DR

This paper proves that in two-dimensional low-temperature Ising models, starting from a biased disordered state leads to rapid convergence to the plus phase, extending previous zero-temperature results.

Contribution

It establishes rapid phase ordering for low-temperature 2D Ising dynamics from biased initial states, using a novel spacetime multiscale coupling method.

Findings

Rapid convergence to plus phase from biased initial states at low temperature.

Extension of zero-temperature absorption results to the entire low-temperature regime.

Development of a spacetime multiscale coupling technique for Ising dynamics.

Abstract

We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 <1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p>p_0$, run at any low temperature $\beta>\beta_c$ converges rapidly to the plus phase measure $\pi^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$\beta$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.