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)=P p{# W=∞}. Some critical values ofp,p H andp T, are defined, respectively, as the smallest value ofp for whichθ(p)> 0, and for which the expectation of #W is infinite. Formally,p H=inf {p∶θ(p)>0} andp T=inf{p∶ E p{#W}=∞}. We show for fairly general graphsGthat ifp <p T, thenP P{#W ⩾ n} decreases exponentially inn. For the special casesG =G 0= the simple quadratic lattice andG 1= the graph which corresponds to bond-percolation on ℤ2, we obtain upper and lower bounds forθ(p) of the formC¦p¦-P H¦α, and bounds forEp{#W} of the formC¦p−p H¦−α. We also investigate smoothness properties of Δ(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 Δ(·) has exactly one singularity, namely, atp=p H. For the graphsG 0 andG 1, (i.e., site or bond percolation on ℤ2) we show that Δ(p) is analytic atp ≠ p H and has two continuous derivatives atp=p H. The emphasis is on rigorous proofs.
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Research supported by the NSF through a grant to Cornell University.
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Kesten, H. Analyticity properties and power law estimates of functions in percolation theory. J Stat Phys 25, 717–756 (1981). https://doi.org/10.1007/BF01022364
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DOI: https://doi.org/10.1007/BF01022364