The Percolation Transition in the Zero-Temperature Domany Model

Abstract

We analyze a deterministic cellular automaton ${\sigma }^{\cdot }=({\sigma }^n:n\ge 0)$ corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{H}$. The state space ${\mathcal{S}}_{\mathbb{H}}=\{-1,+1{\}}^{\mathbb{H}}$ 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 $\mathcal{A}$ 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 ${\sigma }_x^{\cdot }$'s with $x\in \mathcal{A}$ (respectively, $\mathcal{B}$) are updated simultaneously at odd (resp., even) times, making ${\sigma }_x^{\cdot }$ 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).

12 pages, 1 figure

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