Large clusters in a correlated percolation model

TL;DR

This paper investigates a correlated site percolation model on a cubic lattice, analyzing the size and fractal dimensions of large clusters near the percolation threshold, revealing a specific scaling law.

Contribution

It introduces a detailed analysis of cluster sizes and fractal dimensions in a correlated percolation model, providing new scaling relations and insights into cluster structure.

Findings

Cluster mass scales as L^{5/2}/r^{5/6} for large L and r

Fractal dimension of clusters is 5/2 for all r

Percolation transition occurs at u_c=3.15

Abstract

We consider a correlated site percolation problem on a cubic lattice of size $L^3$, with $16\le L\le 512$. The sites of an initially full lattice are removed by a random walk of ${\cal N}=uL^3$ steps. When the parameter $u$ crosses a threshold $u_c=3.15$, a large system transitions between percolating and non-percolating states. We study the $L$-dependence of the mean mass (number of sites) $M_r$ of the $r$th largest cluster, as well as $r$-dependence of $M_r$ for various system sizes $L$ at $u_c$. We demonstrate that $M_r\sim L^{5/2}/r^{5/6}$ for moderate or large $L$ and $r\gg 1$, and also conclude that for {\em any} $r$ the fractal dimensions of the clusters are $5/2$.