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October, 1987 Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis
J. T. Chayes, L. Chayes, C. M. Newman
Ann. Probab. 15(4): 1272-1287 (October, 1987). DOI: 10.1214/aop/1176991976


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) $\leq \exp(-\lbrack c/\log n\rbrack n^{(d-1)/d})$. 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 $p_c$.


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J. T. Chayes. L. Chayes. C. M. Newman. "Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis." Ann. Probab. 15 (4) 1272 - 1287, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0627.60099
MathSciNet: MR905331
Digital Object Identifier: 10.1214/aop/1176991976

Primary: 60K35
Secondary: 60D05

Keywords: Bernoulli percolation , cluster size distribution , Invasion percolation , truncated connectivity function

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 4 • October, 1987
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