The Percolation Transition in the Zero-Temperature Domany Model


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Abstract

We analyze a deterministic cellular automaton σ=(σn:n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice ℍ. The state space 𝒮 = {−1, +1} consists of assignments of −1 or +1 to each site of ℍ and the initial state σ0 = {σ0x}x ε ℍ is chosen randomly with P0x = +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 σx's with x ε 𝒜 (respectively, ℬ) are updated simultaneously at odd (resp., even) times, making σx 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).

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