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 sigma(t), in {-1, + 1}(H), updates each site v asynchronously in continuous time t greater than or equal to 0, so that sigma(v)(t) agrees with a majority of its (three) neighbors. The initial sigma(v) (0) s are i. i. d. with P [sigma(v) (0) = + 1] = p is an element of [0, 1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t --> infinity and p --> 1/2. Denoting by chi(+) (t, p) the expected size of the plus cluster containing the origin, we (1) prove that chi(+) (infinity, 1/ 2) = infinity and (2) study numerically critical exponents associated with the divergence of chi(+) (infinity, p) as p up arrow1/ 2. A detailed finite-size scaling analysis suggests that the exponents gamma and v of this t= infinity (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 sigma(t) --> sigma(infinity) as t --> infinity is exponential.