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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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