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 $c = c(p,d)$.