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Bernoulli percolation above threshold: An invasion percolation analysis. (English) Zbl 0627.60099
Using the invasion percolation process, we prove the following for Bernoulli percolation on \({\mathbb{Z}}^ d\) \((d>2):\)
(1) exponential decay of the truncated connectivity, \(\tau '_{xy}\equiv P(x\) and y belong to the same finite cluster)\(\leq \exp (-m\| x- y\|)\); (2) infinite differentiability of \(P_{\infty}(p)\), the infinite cluster density, and of \(\chi\) ’(p), the expected size of finite clusters, as functions of p, the density of occupied bonds; and (3) upper bounds on the cluster size distribution tail, \(P_ n\equiv P(the\) cluster of the origin contains exactly n bonds)\(\leq \exp (-[c/\log n]n^{(d-1)/d}).\)
Such results (without the log n denominator in (3)) were previously known for \(d=2\) and \(p>p_ c\), the usual percolation threshold, or for \(d>2\) and p close to 1. We establish these results for all \(d>2\) when p is above a limit of “slab thresholds”, conjectured to coincide with \(p_ c\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
82B43 Percolation
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