The Annals of Probability

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

Kenneth S. Alexander

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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


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