Cluster size and boundary distribution near percolation threshold

It is shown that the shape of the large, random clusters, near the critical percolation concentration c₀, 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 〈b〉/〈n〉 is simply related to the critical concentration c₀. The detailed results of a Monte Carlo calculation, previously reported, are given for c<c₀ on a simple square lattice; they yield an empirical formula for the probability distribution P(n,b), for finding a cluster of size n and boundary b, that is proportional to a Gaussian in b/n, which is independent of concentration and which narrows to a δ function at b/n=α₀, n→∞. The asymptotic behavior of the Gaussian form gives the critical exponents β=0.19±0.16, and γ=2.34±0.3, and α₀, gives the critical concentration c₀=0.587±0.14, in agreement with previous determinations.