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, 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 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.

Highlights

  • Consider critical site percolation on Zd with d ≥ 2

  • Most of the work in our proof concerns a ‘construction’ which reduces the problem to a topological one

  • This is solved by applying a topological fact, Lemma 2.12 below, which follows from Brouwer’s fixed point theorem

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Summary

Introduction and statement of the main results

Consider site percolation on Zd with d ≥ 2. (ii) The precise factor 2 in the expression Λ(2n) in (1.3) is not essential for Cerf’s applications mentioned in Remark 1.4(ii): a bigger constant, which may even depend on the dimension, would work as well (with tiny, straightforward modifications of Cerf’s arguments). We will prove that, roughly speaking, each point is the sum of ‘just’ d good points, which together with the previous statement gives the exponent d2 in Theorem 1.1 To prove this we first show, again using Corollary 2.3 (but with general Γ, boxes) the existence of suitable paths of good points and turn the problem into a topological issue. This reformulation is less ‘compact’ than that of Theorem 1.1 but has the advantage that it is more natural with respect to the approach in our proof (where, as we will see, we first fix an n and distinguish between ‘good’ and other points in Λ(n))

Preliminaries
The set of ‘good’ vertices and its properties

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