The Percolation Transition in the Zero-Temperature Domany Model

Abstract

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