### Abstract

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 σ^{0} = {σ_{x}
^{0}} _{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 σ^{n}, 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 σ^{n}, n ∈ [1, ∞], have the same values as for standard two dimensional independent site percolation (on the triangular lattice).

Original language | English (US) |
---|---|

Pages (from-to) | 1199-1210 |

Number of pages | 12 |

Journal | Journal of Statistical Physics |

Volume | 114 |

Issue number | 5-6 |

State | Published - Mar 2004 |

### Fingerprint

### Keywords

- Cellular automaton
- Critical exponents
- Dependent percolation
- University
- Zero-temperature dynamics

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*114*(5-6), 1199-1210.

**The percolation transition in the zero-temperature Domany model.** / Camia, Federico; Newman, Charles.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 114, no. 5-6, pp. 1199-1210.

}

TY - JOUR

T1 - The percolation transition in the zero-temperature Domany model

AU - Camia, Federico

AU - Newman, Charles

PY - 2004/3

Y1 - 2004/3

N2 - We analyze a deter ministic 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 script capital L signℍ = {-1, +1} ℍ consists of assignments of -1 or +1 to each site of ℍ and the initial state σ0 = {σx 0} x ∈ ℍ is chosen randomly with P(σx 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 σx .'s with x ∈ script A sign (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).

AB - We analyze a deter ministic 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 script capital L signℍ = {-1, +1} ℍ consists of assignments of -1 or +1 to each site of ℍ and the initial state σ0 = {σx 0} x ∈ ℍ is chosen randomly with P(σx 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 σx .'s with x ∈ script A sign (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).

KW - Cellular automaton

KW - Critical exponents

KW - Dependent percolation

KW - University

KW - Zero-temperature dynamics

UR - http://www.scopus.com/inward/record.url?scp=3543046778&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3543046778&partnerID=8YFLogxK

M3 - Article

VL - 114

SP - 1199

EP - 1210

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -