TY - JOUR

T1 - A lower bound for point-to-point connection probabilities in critical percolation

AU - Van Den Berg, J.

AU - Don, H.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Connection probabilities

KW - Critical percolation

UR - http://www.scopus.com/inward/record.url?scp=85090530166&partnerID=8YFLogxK

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U2 - 10.1214/20-ECP326

DO - 10.1214/20-ECP326

M3 - Article

AN - SCOPUS:85090530166

SN - 1083-589X

VL - 25

SP - 1

EP - 9

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

M1 - 47

ER -