Probability distribution and sizes of spanning clusters at the percolation thresholds

TL;DR

This paper investigates the probability distribution and size variation of spanning clusters at the percolation threshold, revealing an exponential decay in cluster count probability and a scaling relation for cluster sizes.

Contribution

It provides large-scale simulation results showing the distribution of spanning clusters follows an exponential decay and analyzes how average cluster size varies with the number of spanning clusters.

Findings

Probability distribution P(n) ~ exp(-a n^2) for spanning clusters.

Average size of spanning clusters scales as L^D with D = d - β/ν.

Average size decreases smoothly as the number of spanning clusters increases.

Abstract

For random percolation at p_c, the probability distribution P(n) of the number of spanning clusters (n) has been studied in large scale simulations. The results are compatible with $P(n) \sim \exp(-an^2)$ for all dimensions. We also study the variation of the average size (mass) of the spanning clusters when there are more than one spanning cluster. While the average size of the spanning clusters scales as usual with $L^D$ where $D = d- \beta/\nu$ for any number of clusters, it shows a smooth decrease as the number of spanning clusters increases.