(227d) Monte Carlo Simulation of Percolation Properties, Including Cluster Numbers and Elastic Backbones

We describe various simulation procedures to simulate properties in percolation, including the Leath Cluster Growth method, the Hoshen-Kopelman method, and the Newman-Ziff method. We first give an introduction and overview into percolation including its relation to the Potts model and thermodynamic systems. Then we discuss specific problems studied using these Monte Carlo methods, including crossing probability and cluster density, for systems at and near the critical point. We also talk about the importance of the geometric embedding and show how in two dimensions the isoradial construction can be used to find the correct embedding to have isotropic growth. These embeddings are important in practical problems to get a correct understanding of the anisotropy of the flow, especially for problems where there is anisotropic bond probabilities.