The ramification of large clusters near percolation threshold

A general upper bound to the perimeter polynomial is demonstrated for the site percolation problem and it is concluded that at percolation threshold p=pc the limiting mean perimeter-to-size ratio of (1-pc)/pc exists for large clusters. The standard deviation of the perimeter-to-size distribution approaches zero as n-12/. These properties are illustrated by an exact solution of the percolation problem on the expanded cactus. The connection between the asymptotic form of the perimeter polynomial and critical exponents are discussed and three recent suggestions for this asymptotic form are compared.