Order propagation near the percolation threshold

Abstract

A new Monte Carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold. The bonds in each cluster are partitioned into three distinct connectivity classes, 'red' (singly connected backbone bonds), 'blue' (multiply connected backbone bonds) and 'yellow' (non-backbone bonds, often called dangling ends). Among the new cluster properties studied are the mean number of red bonds, a critical quantity diverging at p_c with exponent y_R approximately ≃ 1, and the length of the shortest connected path through the cluster which is critical with exponent y_(min)=1.35 ± 0.02. For all cluster properties studies, the authors also compute averages over only the largest clusters; the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes.