L. R. Fontes, R. H. Schonmann and V. Sidoravicius
Stretched exponential fixation in stochastic Ising models at zero temperature
(364K, postcript)
ABSTRACT. We study a class of continuous time Markov processes, which describes
$\pm 1$ spin flip dynamics on the hypercubic lattice $Z^d, d \ge 2$,
with initial spin configurations chosen according to the Bernoulli product
measure with density $p$ of spins $+1$.
During the evolution the spin at each site flips at rate $c = 0,$ or
$0< \alpha \le 1$, or $1$, depending on whether, respectively, a majority
of spins of nearest neighbors to this site exists and agrees with the
value of the spin at the given site, or does not exist (there is a tie),
or exists and disagrees with the value of the spin at the given site.
These dynamics correspond to various stochastic Ising models at 0
temperature, for the Hamiltonian with uniform ferromagnetic interaction
between nearest neighbors. In case $\alpha =1$, the dynamics is also a
threshold voter model. We show that if $p$ is sufficiently
close to $1$, then the system fixates in the sense that for almost every
realization of the initial configuration and dynamical evolution, each site
flips only finitely many times, reaching eventually the state $+1$.
Moreover, we show that in this case the probability $q(t)$ that a given
spin is in state $-1$ at time $t$ satisfies the bound: for arbitrary
$\epsilon > 0$, $q(t) \le \exp(-t^{(1/d) - \epsilon})$, for large $t$.
In $d=2$ we obtain the complementary bound: for arbitrary
$\epsilon > 0$, $q(t) \ge \exp(-t^{(1/2) + \epsilon})$, for large $t$.