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
SN - 0022-4715
VL - 111
SP - 57
EP - 72
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 1-2
ER -