A lower bound for point-to-point connection probabilities in critical percolation

Abstract

Consider critical site percolation on $\mathbb{Z} ^{d}$ with $d \geq 2$. We prove a lower bound of order $n^{- d^{2}}$ for point-to-point connection probabilities, where $n$ is the distance between the points.

Most of the work in our proof concerns a ‘construction’ which finally reduces the problem to a topological one. This is then solved by applying a topological fact, Lemma 2.12 below, which follows from Brouwer’s fixed point theorem.

Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 [1] to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.