Distribution of Dangling Ends on the Incipient Percolation Cluster

TL;DR

This paper investigates the distribution of dangling end sizes on the incipient percolation cluster, revealing a power-law decay with exponents linked to the cluster's fractal dimensions, supported by numerical and scaling analyses.

Contribution

It introduces a theoretical and numerical analysis of dangling end size distribution on percolation clusters, connecting the decay exponent to fractal dimensions.

Findings

Power-law decay of dangling end distribution with exponent k=0.83 in 2D

Exponent k=0.74 in 3D matches theoretical predictions

Good agreement between numerical results and scaling theory

Abstract

We study numerically and by scaling arguments the probability P(M)dM that a given dangling end of the incipient percolation cluster has a mass between M and M + dM. We find by scaling arguments that P(M) decays with a power law, P(M)~M^(-(1+k)), with an exponent k=dBf/df, where df and dBf are the fractal dimensions of the cluster and its backbone, respectively. Our numerical results yield k=0.83 in d=2 and k=0.74 in d=3 in very good agreement with theory.