Cluster statistics of the lattice gas model in three and two dimensions

Cluster statistics in the lattice gas system were evaluated in three dimensions (d = 3) below the critical point, and in two dimensions (d = 2) above Tc using a Monte Carlo method. Defining clusters as sets of l particles connected by nearest-neighbor bonds, we found remarkable deviation from semiphenornenologic cluster probability formulas below Tc. This deviation is attributed to the formation of spongelike noncompact macroclusters near the percolation temperature Tp < Tc. Above T c the cluster probabilities in two dimensions may be scaled in the variable l̄ = (1 - Tc/T) lσ with σ = 0.53 below Tc. In contrast to the two-dimensional case of T < T c here the cluster formulas cannot explain the distribution up to clusters with 1 ≤ 2000 particles. For T→∞ (d = 2) the cluster probability pl decays as pl∼exp(-const 1 ζ) with (ζ ≈ 1 for sufficiently large l. This supports recent arguments that the Griffiths singularity for dilute systems is an essential one. Copyright © 1976 American Institute of Physics.