TY - JOUR

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

AU - Douglas Howard, C.

AU - Newman, Charles M.

N1 - Funding Information:
We thank Alan Sokal for very useful discussions and advice about finite-size scaling. We also note the participation of Elizabeth Zollinger as part of an undergraduate research experience supported by the NSF VIGRE program. Research of C.D.H. supported in part by NSF Grants DMS-98-15226 and DMS-02-03943 and by a Eugene Lang Research Fellowship. Research of C.M.N. supported in part by NSF Grants DMS-98-03267 and DMS-01-04278.

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

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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 -