## The Annals of Probability

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

Kenneth S. Alexander

#### 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

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