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 bn, which is independent of concentration and which narrows to a function at bn=₀, n. The asymptotic behavior of the Gaussian form gives the critical exponents =0. 190. 16, and =2. 340. 3, and ₀, gives the critical concentration c₀=0. 5870. 14, in agreement with previous determinations.