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A lower bound for point-to-point connection probabilities in critical percolation. (English) Zbl 1456.60266
Summary: 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 R. Cerf in [Ann. Probab. 43, No. 5, 2458–2480 (2015; Zbl 1356.60163)] 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.
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
Full Text: DOI Euclid
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