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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.
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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References:
[1] Raphaël Cerf, A lower bound on the two-arms exponent for critical percolation on the lattice, Ann. Probab. 43 (2015), no. 5, 2458-2480. · Zbl 1356.60163
[2] Hugo Duminil-Copin and Vincent Tassion, A new proof of the sharpness of the phase transition for Bernoulli percolation on \(\mathbb{Z}^d \), Enseign. Math. 62 (2016), no. 1-2, 199-206. · Zbl 1359.60118
[3] Robert Fitzner and Remco van der Hofstad, Mean-field behavior for nearest-neighbor percolation in \(d>10\), Electron. J. Probab. 22 (2017), Paper No. 43, 65. · Zbl 1364.60130
[4] Andrzej Granas and James Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. · Zbl 1025.47002
[5] J. M. Hammersley, Percolation processes: Lower bounds for the critical probability, Ann. Math. Statist. 28 (1957), 790-795. · Zbl 0091.13903
[6] Takashi Hara, Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals, Ann. Probab. 36 (2008), no. 2, 530-593. · Zbl 1142.82006
[7] Markus Heydenreich and Remco van der Hofstad, Progress in high-dimensional percolation and random graphs, CRM Short Courses, Springer, 2017. · Zbl 1445.60003
[8] Naeem Jan and Dietrich Stauffer, Random site percolation in three dimensions, International Journal of Modern Physics C 09 (1998), no. 02, 341-347.
[9] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, One-arm exponent for critical 2D percolation, Electron. J. Probab. 7 (2002), no. 2, 13. · Zbl 1015.60091
[10] Carlo Miranda, Un’osservazione su un teorema di Brouwer, Boll.
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