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Power-law bounds for critical long-range percolation below the upper-critical dimension

Hutchcroft, Tom (2020) Power-law bounds for critical long-range percolation below the upper-critical dimension. . (Unpublished)

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We study long-range Bernoulli percolation on ℤd in which each two vertices x and y are connected by an edge with probability 1−exp(−β‖x−y‖−d−α). It is a theorem of Noam Berger (CMP, 2002) that if 0<α<d then there is no infinite cluster at the critical parameter βc. We give a new, quantitative proof of this theorem establishing the power-law upper bound Pβc(|K|≥n)≤Cn−(d−α)/(2d+α) for every n≥1, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality (2−η)(δ+1)≤d(δ−1) relating the cluster-volume exponent δ and two-point function exponent η.

Item Type:Report or Paper (Discussion Paper)
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Hutchcroft, Tom0000-0003-0061-593X
Additional Information:Dedicated to Harry Kesten, November 19, 1931 - March 29, 2019. We thank Jonathan Hermon for his careful reading of an earlier version of this manuscript, and thank Gordon Slade for helpful discussions on the physics literature. We also thank the anonymous referee for their helpful comments and corrections.
Record Number:CaltechAUTHORS:20210924-202126385
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111034
Deposited By: George Porter
Deposited On:27 Sep 2021 14:55
Last Modified:27 Sep 2021 14:55

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