The Percolation Transition in the Zero-Temperature Domany Model

Abstract

We analyze a deterministic cellular automaton σ ^⋅=(σ ⁿ:n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{N}$. The state space $\mathcal{S}_\mathbb{H} = \left\{ { - 1, + 1} \right\}^\mathbb{H}$ consists of assignments of −1 or +1 to each site of $\mathbb{H}$ and the initial state $\sigma ^0 = \left\{ {\sigma _{^x }^0 } \right\}_{x \in \mathbb{H}}$ is chosen randomly with P(σ ⁰ _x=+1)=p∈[0,1]. The sites of $\mathbb{H}$ are partitioned in two sets $\mathcal{A}$ and $\mathcal{B}$ so that all the neighbors of a site x in $\mathcal{A}$ belong to $\mathcal{B}$ and vice versa, and the discrete time dynamics is such that the σ ^⋅ _x 's with ${x \in \mathcal{A}}$ (respectively, $\mathcal{B}$) are updated simultaneously at odd (resp., even) times, making σ ^⋅ _x agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p=1/2 in the percolation models defined by σ ⁿ, for all times n∈[1,∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν, and η of the dependent percolation models defined by σ ⁿ, n∈[1,∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).