## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 4 (1990), 1547-1562.

### Lower Bounds on the Connectivity Function in all Directions for Bernoulli Percolation in Two and Three Dimensions

#### Abstract

The probability $P\lbrack 0 \leftrightarrow x \rbrack$ of connection of 0 to $x$ by a path of occupied bonds for Bernoulli percolation at density $p$ below the critical point is known to decay exponentially for each direction $x \in \mathbb{Z}^d$, in that $P\lbrack 0 \leftrightarrow nx \rbrack \approx e^{-n\sigma g(x)}$ as $n \rightarrow \infty$ for some $\sigma > 0$ and $g(x)$ of order $\|x\|$. This approximation is also an upper bound: $P\lbrack 0 \leftrightarrow x \rbrack \leq e^{-\sigma g(x)}$ for all $x$. Here a complementary power-law lower bound is established for $d = 2$ and 3: $P\lbrack 0 \leftrightarrow x \rbrack \geq c\|x\|^{-r} e^{-\sigma g(x)}$ for some $r = r(d)$ and $c = c(p,d)$.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 4 (1990), 1547-1562.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990631

**Digital Object Identifier**

doi:10.1214/aop/1176990631

**Mathematical Reviews number (MathSciNet)**

MR1071808

**Zentralblatt MATH identifier**

0718.60110

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 82A43

**Keywords**

Percolation connectivity function Ornstein-Zernike behavior powerlaw correction

#### Citation

Alexander, Kenneth S. Lower Bounds on the Connectivity Function in all Directions for Bernoulli Percolation in Two and Three Dimensions. Ann. Probab. 18 (1990), no. 4, 1547--1562. doi:10.1214/aop/1176990631. https://projecteuclid.org/euclid.aop/1176990631