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 $\delta$ function as $n\to \infty$; the asymptotic behavior gives $c_0$ and the critical exponents.