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Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jan;89(1):012120. Epub 2014 Jan 15.

Geometric structure of percolation clusters.

Author information

1
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
2
School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.

Abstract

We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).

PMID:
24580185
DOI:
10.1103/PhysRevE.89.012120
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