Cluster size and boundary distribution near percolation threshold

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 $c<\mathrm{}{c}_0$ 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 $\delta$ function at $\frac{b}{n}={\alpha }_0$, $n\to \infty$. The asymptotic behavior of the Gaussian form gives the critical exponents $\beta =0.19\mathrm{}\pm 0.16$, and $\gamma =2.34\mathrm{}\pm 0.3$, and ${\alpha }_0$, gives the critical concentration $c_0=0.587\mathrm{}\pm 0.14$, in agreement with previous determinations.