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

TL;DR

This paper establishes a new lower bound for point-to-point connection probabilities in critical percolation on integer lattices, improving previous bounds and connecting topological methods with percolation theory.

Contribution

It introduces a novel lower bound of order n^{-d^2} for connection probabilities, utilizing topological arguments and Brouwer's fixed point theorem.

Findings

Established a lower bound of order n^{-d^2} for connection probabilities.

Improved the exponent in lower bounds compared to previous work.

Connected topological fixed point theorems with percolation probability estimates.

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.