The percolation transition in the zero-temperature Domany model

We analyze a deter ministic cellular automaton σ = (σ^n.: n ≥ 0) corresponding to the zero temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice ℍ The state space script capital L signℍ = {-1, +1} ^ℍ consists of assignments of -1 or +1 to each site of ℍ and the initial state σ⁰ = {σ_x⁰} _{x ∈ ℍ} is chosen randomly with P(σ_x ⁰ = +1) p ∈ [0, 1]. The sites of ℍ are partitioned in two sets script A sign and ℬ so that all the neighbors of a site in script A sign belong to ℬ and vice versa, and the discrete time dynamics is such that the σ_x^.'s with x ∈ script A sign (respectively, 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 σⁿ, 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 σⁿ, n ∈ [1, ∞], have the same values as for standard two dimensional independent site percolation (on the triangular lattice).