Monte Carlo experiments on cluster size distribution in percolation

Abstract

Abstract. Cluster statistics in two- and three-dimensional site percolation problems are derived here by Monte Carlo methods. The average number n, of percolation clusters with s occupied sites each is calculated by up to 19 runs on a 4000 X 4000 triangular lattice near p c. Our data support the two-exponent scaling assumption n, as- ' f ( z ' ) , where z ' = ( p / p c- 1)s". At the percolation threshold p = pc we find for s up to lo6 a rough agreement with the expected power law n, a s- ' over 12 decades in n, ; we can approximate the leading correction term near 5-10 ' by n,cXs-'(l-1.2 s- ~ ' ~ ). If the ratio U, = n, ( p) / n, ( p, ) is plotted against z ' , then all data follow the same curve U, = f ( z ' ) for different p. This scaling function f(z') has a finite slope at z ' = 0, has a maximum f ( z k, =-0.8) = 5 for p below pc, and decays rapidly for z'+*m. For 5-+m at fixed p this rapid decay corresponds to In n, Cc-s" ' above p c and In n, a--s below p c. Apart from finite-size corrections we find the second moment x = Z s 2 n, diverges as 1 p-pC(-', with y = 2.4, on both sides of the phase transition; the amplitude ratio x ( p < p C) / x ( p> p, ) is about 200. The fraction of occupied sites belonging to the infinite cluster vanishes as ( p-p,)'. with p-0.13. In three dimen-sions using system sizes up to 400 x 400 x 400 the two-exponent scaling function is also supported, with the same universal function f ( z ' ) valid for both the simple cubic and BCC lattices. f ( z ' ) has a maximum f ( z; , =-0.8) = 1.6. The amplitude ratio is approximately 11. Our conclusions are in general consistent with but more complete than other recent