Journal of Statistical Physics, Vol. 66, Nos. 3/4, 1992, p. 1123-1138.
The cluster density function of independent percolation k in a d-dimensional lattice is considered. For each n, it is shown that ~c(p) has finite nth left-derivative at critical probability Pc if d is sufficiently large. This result agrees with the Bethe lattice approximation, where the n th one-sided derivative of K(p) is bounded at Pc for all n.