Exact Enumeration and Scaling for Fragmentation of PercolationClusters
Bulletin of the American Physical Society
The fragmentation properties of percolation clusters yield information about their structure. Monte Carlo simulations and exact cluster enumeration for a square bond lattice and exact calculations for the Bethe lattice are used to study the fragmentation probability as(p) of clusters of mass s at an occupation probability p and the likelihood bs's(p) that fragmentation of an s cluster will result in a daughter cluster of mass s′. Evidence is presented to support the scaling laws as(pc)∼s and bs's(pc)=s−φg(s′/s), with φ=2-σ given by the standard cluster-number scaling exponent σ. Simulations for d=2 verify the finite-size-scaling form cs'sL(pc)=s1−φg̃(s′/s,s/Ldf) of the product cs's(pc)=as(pc)bs's(pc), where L is the lattice size and df is the fractal dimension. Exact calculations of the fragmentation probability fst of a cluster of mass s and perimeter t indicate that branches are important even on the maximum perimeter clusters. These calculations also show that the minimum of bs's(p) near s′=s/2, where the two daughter masses are comparable, deepens with increasing p.
"Exact Enumeration and Scaling for Fragmentation of Percolation Clusters," B. F. Edwards, M. F. Gyure, and M. V. Ferer, Bull. Am. Phys. Soc. 37, 687 (1992).