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2020 A lower bound for point-to-point connection probabilities in critical percolation
J. van den Berg, H. Don
Electron. Commun. Probab. 25: 1-9 (2020). DOI: 10.1214/20-ECP326


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.


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J. van den Berg. H. Don. "A lower bound for point-to-point connection probabilities in critical percolation." Electron. Commun. Probab. 25 1 - 9, 2020.


Received: 3 March 2020; Accepted: 9 June 2020; Published: 2020
First available in Project Euclid: 1 July 2020

zbMATH: 07225540
MathSciNet: MR4125794
Digital Object Identifier: 10.1214/20-ECP326

Primary: 60K35
Secondary: 82B43

Keywords: connection probabilities , Critical percolation


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