A Caltech Library Service

On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition

Hutchcroft, Tom (2022) On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition. Journal of Statistical Physics, 189 (1). Art. No. 6. ISSN 0022-4715. doi:10.1007/s10955-022-02967-7.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


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 ∇pc 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↑pc, as is expected to occur in percolation on Zd 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:Article
Related URLs:
URLURL TypeDescription ItemDiscussion Paper
Hutchcroft, Tom0000-0003-0061-593X
Additional Information:This work was carried out while the author was a Senior Research Associate at the University of Cambridge, and 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.
Funding AgencyGrant Number
European Research Council (ERC)804166
Issue or Number:1
Record Number:CaltechAUTHORS:20220823-628154700
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:116429
Deposited By: Melissa Ray
Deposited On:30 Aug 2022 20:27
Last Modified:30 Aug 2022 20:27

Repository Staff Only: item control page