The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

Abstract

On the planar hexagonal lattice $\mathbb{H}$, we analyze the Markov process whose state σ(t), in $\{ - 1, + 1\} ^\mathbb{H}$, updates each site v asynchronously in continuous time t≥0, so that σ _v (t) agrees with a majority of its (three) neighbors. The initial σ _v (0)'s are i.i.d. with P[σ _v (0)=+1]=p∈[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t→∞ and p→1/2. Denoting by χ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that χ⁺(∞,1/2)=∞ and (2) study numerically critical exponents associated with the divergence of χ⁺(∞,p) as p↑1/2. A detailed finite-size scaling analysis suggests that the exponents γ and ν of this t=∞ (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which σ(t)→σ(∞) as t→∞ is exponential.