### Abstract

On the planar hexagonal lattice ℍ, we analyze the Markov process whose state σ(t), in {-1,+1} ^{ℍ}, updates each site v asynchronously in continuous time t ≥ 0, so that σ _{v},(t) agrees with a majority of its (three) neighbors. The initial σ _{v}(0)'s are i.i.d. with P[σ _{v}(0) = +1] = p ∈ [0, 1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t → ∞ and p → 1/2. Denoting by x ^{+}(t, p) the expected size of the plus cluster containing the origin, we (1) prove that χ ^{+}(∞, 1/2) = ∞ and (2) study numerically critical exponents associated with the divergence of %chi; ^{+}(∞, p) as p ↑ 1/2. A detailed finite-size scaling analysis suggests that the exponents γ and ν of this t = ∞ (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which σ(t) → σ(∞) as t → ∞ is exponential.

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

Pages (from-to) | 57-72 |

Number of pages | 16 |

Journal | Journal of Statistical Physics |

Volume | 111 |

Issue number | 1-2 |

DOIs | |

State | Published - Apr 2003 |

### Fingerprint

### Keywords

- Critical exponents
- Dependent percolation
- Glauber dynamics
- Hexagonal lattice
- Ising spin dynamics

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*111*(1-2), 57-72. https://doi.org/10.1023/A:1022296706006

**The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice.** / Douglas Howard, C.; Newman, Charles.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 111, no. 1-2, pp. 57-72. https://doi.org/10.1023/A:1022296706006

}

TY - JOUR

T1 - The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

AU - Douglas Howard, C.

AU - Newman, Charles

PY - 2003/4

Y1 - 2003/4

N2 - On the planar hexagonal lattice ℍ, we analyze the Markov process whose state σ(t), in {-1,+1} ℍ, updates each site v asynchronously in continuous time t ≥ 0, so that σ v,(t) agrees with a majority of its (three) neighbors. The initial σ v(0)'s are i.i.d. with P[σ v(0) = +1] = p ∈ [0, 1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t → ∞ and p → 1/2. Denoting by x +(t, p) the expected size of the plus cluster containing the origin, we (1) prove that χ +(∞, 1/2) = ∞ and (2) study numerically critical exponents associated with the divergence of %chi; +(∞, p) as p ↑ 1/2. A detailed finite-size scaling analysis suggests that the exponents γ and ν of this t = ∞ (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which σ(t) → σ(∞) as t → ∞ is exponential.

AB - On the planar hexagonal lattice ℍ, we analyze the Markov process whose state σ(t), in {-1,+1} ℍ, updates each site v asynchronously in continuous time t ≥ 0, so that σ v,(t) agrees with a majority of its (three) neighbors. The initial σ v(0)'s are i.i.d. with P[σ v(0) = +1] = p ∈ [0, 1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t → ∞ and p → 1/2. Denoting by x +(t, p) the expected size of the plus cluster containing the origin, we (1) prove that χ +(∞, 1/2) = ∞ and (2) study numerically critical exponents associated with the divergence of %chi; +(∞, p) as p ↑ 1/2. A detailed finite-size scaling analysis suggests that the exponents γ and ν of this t = ∞ (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which σ(t) → σ(∞) as t → ∞ is exponential.

KW - Critical exponents

KW - Dependent percolation

KW - Glauber dynamics

KW - Hexagonal lattice

KW - Ising spin dynamics

UR - http://www.scopus.com/inward/record.url?scp=0346492931&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346492931&partnerID=8YFLogxK

U2 - 10.1023/A:1022296706006

DO - 10.1023/A:1022296706006

M3 - Article

AN - SCOPUS:0346492931

VL - 111

SP - 57

EP - 72

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -