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 (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400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) | |||||||||
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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. | |||||||||
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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: | 30 Aug 2022 20:25 |
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