Cluster size and boundary distribution near percolation threshold

Abstract

It is shown that the shape of the large, random clusters, near the critical percolation concentration ${c}_{0}$, is such that their mean boundary $〈b〉$ is proportional to their mean bulk $〈n〉$ and this is illustrated by an argument which shows that the dimension of the boundary is the same as that of the bulk. The resulting ratio $\frac{〈b〉}{〈n〉}$ is simply related to the critical concentration ${c}_{0}$. The detailed results of a Monte Carlo calculation, previously reported, are given for $c<{c}_{0}$ on a simple square lattice; they yield an empirical formula for the probability distribution $\mathcal{P}(n,b)$, for finding a cluster of size $n$ and boundary $b$, that is proportional to a Gaussian in $\frac{b}{n}$, which is independent of concentration and which narrows to a $\ensuremath{\delta}$ function at $\frac{b}{n}={\ensuremath{\alpha}}_{0}$, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic behavior of the Gaussian form gives the critical exponents $\ensuremath{\beta}=0.19\ifmmode\pm\else\textpm\fi{}0.16$, and $\ensuremath{\gamma}=2.34\ifmmode\pm\else\textpm\fi{}0.3$, and ${\ensuremath{\alpha}}_{0}$, gives the critical concentration ${c}_{0}=0.587\ifmmode\pm\else\textpm\fi{}0.14$, in agreement with previous determinations.