Analyticity properties and power law estimates of functions in percolation theory

Abstract

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 probabilityθ is the probability that #W=∞, i.e.,θ(p)=Pp{# W=∞}. Some critical values ofp,pH andpT, are defined, respectively, as the smallest value ofp for whichθ(p)> 0, and for which the expectation of #W is infinite. Formally,pH=inf {p∶θ(p)>0} andpT=inf{p∶ Ep{#W}=∞}. We show for fairly general graphsGthat ifp <pT, thenPP{#W ⩾ n} decreases exponentially inn. For the special casesG =G0= the simple quadratic lattice andG1= the graph which corresponds to bond-percolation on ℤ2, we obtain upper and lower bounds forθ(p) of the formC¦p¦-PH¦α, and bounds forEp{#W} of the formC¦p−pH¦−α. We also investigate smoothness properties of Δ(p)=Ep{number of clusters per site} =Ep {(#W)−1; (#W) ⩾ 1}. This function was introduced by Sykes and Essam, who assumed that Δ(·) has exactly one singularity, namely, atp=pH. For the graphsG0 andG1, (i.e., site or bond percolation on ℤ2) we show that Δ(p) is analytic atp ≠ pH and has two continuous derivatives atp=pH. The emphasis is on rigorous proofs.