Exact distribution of cluster size and perimeter for two-dimensional percolation

Exact series expansion data of Sykes et al. are used to calculate the average numberc_n and perimeters_n of clusters of sizen20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp_n we find a sharply peaked distribution of perimeterss_n with mean s_n=((1–p_n)/p_c)n+O(n) and width s_n²–S_n²n^1.6 where1/=0.39. This perimeter s_n should not be interpreted as a cluster surface in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc_n withn is consistent with the postulated asymmetry aboutp_c: logc_n–n forn with1 forp<p_c and1/2 forp>p_c.