To simulate site percolation, the method described in Newman & Ziff (2001) was used. This is a fast tree-based algorithm that builds up sequential occupation while tracking contiguous clusters. Over several runs it generates a micro-canonical ensemble -- the probability of some observable (in this case percolation) as a function of the number of occupied sites. From this the canonical ensemble can be calculated -- the percolation probability as a function of the site occupation probability (independent probability of each site being occupied). Note that the micro-canonical ensemble is discrete and can be stored exactly, whereas the canonical ensemble is continuous. See the paper for details.
There are many possible ways to estimate the percolation threshold. Several are described and analysed for open systems in Ziff & Newman (2002). Here we use a technique that has the best convergence with increasing lattice size of any known estimator. It requires the use of periodic boundary conditions. In this case Machta's method for detecting cluster wrapping (described in Newman & Ziff, 2001) was implemented. This involves keeping track of the offsets of each site from its cluster root; when a new site is adjacent to one in the same cluster, offsets of the two sites are compared. If they differ by more than one lattice point then wrapping has occurred (and they will differ by L).
Cluster wrapping around one axis (horizontal) is measured, ignoring the other. We can then take advantage of a theoretical result giving the probability of this type of cluster wrapping on an infinite lattice at the critical site occupation probability: .
By evaluating p corresponding to this value of R(p) on finite lattices we can estimate pc . This is done with a simple root-finding method: repeated interval bisection until a specified error tolerance is reached. Newman & Ziff (2001) have shown that this estimator has finite-size scaling according to .