## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 4 (1987), 1272-1287.

### Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis

J. T. Chayes, L. Chayes, and C. M. Newman

#### 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 $p_c$.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 4 (1987), 1272-1287.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991976

**Digital Object Identifier**

doi:10.1214/aop/1176991976

**Mathematical Reviews number (MathSciNet)**

MR905331

**Zentralblatt MATH identifier**

0627.60099

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

**Keywords**

Bernoulli percolation invasion percolation truncated connectivity function cluster size distribution

#### Citation

Chayes, J. T.; Chayes, L.; Newman, C. M. Bernoulli Percolation Above Threshold: An Invasion Percolation Analysis. Ann. Probab. 15 (1987), no. 4, 1272--1287. doi:10.1214/aop/1176991976. https://projecteuclid.org/euclid.aop/1176991976