The percolation transition in the zero-temperature Domany model

We analyze a deterministic cellular automaton s ·=(sn: n \ 0) corresponding to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H. The state space SH={-1,+1}H consists of assignments of -1 or +1 to each site of H and the initial state s0={s0 x}x ¥ H is chosen randomly with P(s0 x=+1)=p ¥ [0, 1]. The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the s· x ’s with x ¥A (respectively, B) are updated simultaneously at odd (resp., even) times, making s· 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 sn, for all times n ¥ [1, .]. In this paper, we study the nature of that transition and prove that the critical exponents b, n, and g of the dependent percolation models defined by sn, n ¥ [1, .], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice)