Bond percolation and cluster formation on random lattices

Abstract

We have studied the formation of clusters and the distribution of bonds between sites on random lattices by Monte Carlo and analytic techniques for coordination numbers in the range 3 ≤ z ≤ 12. A comparison between Cayley trees (Bethe lattices) and systems in which closed loops are allowed (cyclic systems) indicates little difference in cluster formation but considerable differences in bond distribution between these two types of lattice. The results suggest that there is little difference between the percolation limits for the two types (at constant z), contrary to the existing results for disordered systems. This work points out a possible weakness in the analytic treatments of the Cayley tree due to the omission of correlation effects, and it also suggests that stochastic treatments of cyclic systems overestimate the critical bond number for percolation.