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Computer Physics Communications

Recent developments in the Monte Carlo approach to percolation problems

Chin‐Kun Hu, Jau-Ann Chen, N. Sh. Izmailian, Peter H. Kleban

Published Apr 1, 2000• 6 citations
Publisher
DOI
10.1016/s0010-4655(99)00238-6

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We present a histogram Monte Carlo method to calculate the existence probability Ep{\mathit{E}}_{\mathit{p}}Ep​, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use Ep{\mathit{E}}_{\mathit{p}}Ep​ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.
The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.