01-287 L. R. Fontes, R. H. Schonmann and V. Sidoravicius

Stretched exponential fixation in stochastic Ising models at zero temperature (364K, postcript) Jul 25, 01

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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 $t$.

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