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

Abstract

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 T_c using a Monte Carlo method. Defining clusters as sets of l particles connected by nearest‐neighbor bonds, we found remarkable deviation from semiphenomenologic cluster probability formulas below T_c. This deviation is attributed to the formation of spongelike noncompact macroclusters near the percolation temperature T_p<T_c. Above T_c the cluster probabilities in two dimensions may be scaled in the variable ?= (1−T_c/T) l^σ with σ=0.53 below T_c. In contrast to the two‐dimensional case of T<T_c here the cluster formulas cannot explain the distribution up to clusters with l?2000 particles. For T→∞ (d=2) the cluster probability p_l decays as p_l∼exp(−constl^ζ) with ζ?1 for sufficiently large l. This supports recent arguments that the Griffiths singularity for dilute systems is an essential one.