Analyticity properties and power law estimates of functions in percolation theory

We consider percolation on the sites of a graphG, e.g., a regulard-dimensional lattice. All sites ofG are occupied (vacant) with probabilityp (respectively,q=1–p), independently of each other.W denotes the cluster of occupied sites containing a fixed site (which will usually be taken to be the origin) andW the cardinality ofW. The percolation probability is the probability that #W=, i.e.,(p)=P_p{# W=}. Some critical values ofp,p_H andp_T, are defined, respectively, as the smallest value ofp for which(p)> 0, and for which the expectation of #W is infinite. Formally,p_H=inf {p(p)>0} andp_T=inf{p E_p{#W}=}. We show for fairly general graphsGthat ifp _T, thenP_P{#W n} decreases exponentially inn. For the special casesG =G₀= the simple quadratic lattice andG₁= the graph which corresponds to bond-percolation on ², we obtain upper and lower bounds for(p) of the formC¦p¦-P_H¦, and bounds forEp{#W} of the formC¦p–p_H¦^–. We also investigate smoothness properties of (p)=E_p{number of clusters per site} =E_p {(#W)^–1; (#W) 1}. This function was introduced by Sykes and Essam, who assumed that (·) has exactly one singularity, namely, atp=p_H. For the graphsG₀ andG₁, (i.e., site or bond percolation on ²) we show that (p) is analytic atp p_H and has two continuous derivatives atp=p_H. The emphasis is on rigorous proofs.Research supported by the NSF through a grant to Cornell University.