Recent developments in the Monte Carlo approach to percolation problems

Abstract

Using a Monte Carlo method, Hu, Lin and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the probability W m for the appearance of m percolating clusters, which implies that the average number of percolating clusters, C, is a universal quantity. Hu and Lin found that C increases linearly with the aspect ratio, R, of the lattice for large R, i.e. in this case C= aR with a constant a. Hu and Lin also found that a is apparently independent of the boundary conditions. For the periodic boundary conditions in both horizontal and vertical directions, Ziff et al. found that the number of clusters per lattice site, n, for percolation on planar lattices of linear dimensions L can be written as n= n c + b/ N, where n c is n in the limit L→∞, b is a constant, and N is the number of lattice sites. Ziff et al. found that b is universal and argued that b is the number of percolating clusters so that its universality may be related to the universality of C. Hu found that for large R, b= b c R, but b c ≠ a. In this paper, we use a cluster Monte Carlo simulation method to calculate the number of clusters per site, n, of a q-state bond-correlated percolation model (QBCPM) which is equivalent to the q-state Potts on L× L square lattices. We find that for q≥2 the slopes of n as a function of 1/ L 2 are negative. For q=2, i.e. the Ising model, we find that our data can be well represented by n=n c−c/L+b/L 2+⋯ , where b can be calculated exactly from conformal field theory, c>0 and can be calculated exactly from the critical internal energy of the Ising model.