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On the derivation of mean-field percolation critical exponents from the triangle condition

Hutchcroft, Tom (2021) On the derivation of mean-field percolation critical exponents from the triangle condition. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400

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Abstract

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram ∇_(p_c) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram ∇^p is unbounded but diverges slowly as p ↑ p_c, as is expected to occur in percolation on ℤ^d at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p↑pc then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2106.06400v2arXivDiscussion Paper
ORCID:
AuthorORCID
Hutchcroft, Tom0000-0003-0061-593X
Additional Information:The author was supported in part by ERC starting grant 804166 (SPRS). We thank Vivek Dewan, Emmanuel Michta, Stephen Muirhead, and Gordon Slade for helpful comments on a previous version of the manuscript.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)804166
Record Number:CaltechAUTHORS:20210924-202147400
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111040
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:27 Sep 2021 16:40
Last Modified:27 Sep 2021 16:40

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