Cluster Shape and Critical Exponents near Percolation Threshold

Abstract

The shape of the large, random clusters, occurring near percolation threshold ${c}_{0}$, is shown to be such that the mean cluster boundary-to-bulk ratio $\frac{〈b〉}{〈n〉}$ gives ${c}_{0}$. A Monte Carlo calculation yields that the cluster size distribution is proportional to a Gaussian in $\frac{b}{n}$ which is independent of concentration and narrows to a $\ensuremath{\delta}$ function as $n\ensuremath{\rightarrow}\ensuremath{\infty}$; the asymptotic behavior gives ${c}_{0}$ and the critical exponents.