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

On the planar hexagonal lattice ℍ, we analyze the Markov process whose state σ(t), in {-1,+1} ^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 x ⁺(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 %chi; ⁺(∞, 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.