Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature

Abstract:

We study a class of continuous time Markov processes, which describes ± 1 spin flip dynamics on the hypercubic latticeℤ^d, d≥ 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 < α≤ 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 α= 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 ɛ > 0, q(t) ≤ exp(−t ^{(1/ d ) −ɛ}), for large t. In d= 2 we obtain the complementary bound: for arbitrary ɛ > 0, q(t) ≥ exp(−t ^{(1/2) +ɛ}), for large t.