Exact enumeration and scaling for fragmentation of percolation clusters.

Abstract

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 ${\mathrm{a}}_{\mathrm{s}}$(p) of clusters of mass s at an occupation probability p and the likelihood ${\mathrm{b}}_{\mathrm{s}\ensuremath{'}\mathrm{s}}$(p) that fragmentation of an s cluster will result in a daughter cluster of mass s\ensuremath{'}. Evidence is presented to support the scaling laws ${\mathrm{a}}_{\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)\ensuremath{\sim}s and ${\mathrm{b}}_{\mathrm{s}\ensuremath{'}\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{s}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}$g(s\ensuremath{'}/s), with \ensuremath{\varphi}=2-\ensuremath{\sigma} given by the standard cluster-number scaling exponent \ensuremath{\sigma}. Simulations for d=2 verify the finite-size-scaling form ${\mathrm{c}}_{\mathrm{s}\ensuremath{'}\mathrm{sL}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{s}}^{1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}$g\ifmmode \tilde{}\else \~{}\fi{}(s\ensuremath{'}/s,s/${\mathrm{L}}^{{\mathrm{d}}_{\mathrm{f}}}$) of the product ${\mathrm{c}}_{\mathrm{s}\ensuremath{'}\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{a}}_{\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)${\mathrm{b}}_{\mathrm{s}\ensuremath{'}\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$), where L is the lattice size and ${\mathrm{d}}_{\mathrm{f}}$ is the fractal dimension. Exact calculations of the fragmentation probability ${\mathrm{f}}_{\mathrm{st}}$ 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 ${\mathrm{b}}_{\mathrm{s}\ensuremath{'}\mathrm{s}}$(p) near s\ensuremath{'}=s/2, where the two daughter masses are comparable, deepens with increasing p.