article · The Annals of Probability · 1987

Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis

Jennifer Chayes, L. Chayes, Charles M. Newman

Cited by 60Open access

Abstract

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 $p_c$.

Cite this paper

APA 7

Chayes, J., Chayes, L., & Newman, C. M. (1987). Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis. The Annals of Probability, 15(4). https://doi.org/10.1214/aop/1176991976

MLA 9

Chayes, Jennifer, et al. “Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis.” The Annals of Probability, vol. 15, no. 4, 1987, https://doi.org/10.1214/aop/1176991976.

Chicago

Chayes, Jennifer, L. Chayes, and Charles M. Newman. 1987. “Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis.” The Annals of Probability 15 (4). https://doi.org/10.1214/aop/1176991976.

Harvard

Chayes, J., Chayes, L. and Newman, C.M. (1987) ‘Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis’, The Annals of Probability, 15(4). Available at: https://doi.org/10.1214/aop/1176991976.

BibTeX

@article{chayes1987bernoulli,
  title = {{Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis}},
  author = {Chayes, Jennifer and Chayes, L. and Newman, Charles M.},
  journal = {The Annals of Probability},
  volume = {15},
  number = {4},
  year = {1987},
  doi = {10.1214/aop/1176991976}
}

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