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

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Consider critical site percolation on Zd with d ≥ 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 2d(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.

Original languageEnglish
Article number47
Pages (from-to)1-9
Number of pages9
JournalElectronic Communications in Probability
Publication statusPublished - 2020


  • Connection probabilities
  • Critical percolation


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