February 2024 Isoperimetric lower bounds for critical exponents for long-range percolation
Johannes Bäumler, Noam Berger
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 721-730 (February 2024). DOI: 10.1214/22-AIHP1342

Abstract

We study independent long-range percolation on Zd where the vertices x and y are connected with probability 1eβxydα for α>0. Provided the critical exponents δ and 2η defined by δ=limnlog(n)log(Pβc(|K0|n)) and 2η=limxlog(Pβc(0x))log(x)+d exist, where K0 is the cluster containing the origin, we show that

δd+(α1)d(α1)and2ηα1.

The lower bound on δ is believed to be sharp for d=1, α[13,1) and for d=2, α[23,1], whereas the lower bound on 2η is sharp for d=1, α(0,1), and for α(0,1] for d>1, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside Zd.

Nous étudions la percolation indépendante de longue portée sur Zd : les sommets x et y sont connectés avec probabilité 1eβxydα pour α>0. En supposant que les exposants critiques δ et 2η définis par δ=limnlog(n)log(Pβc(|K0|n)) et 2η=limxlog(Pβc(0x))log(x)+d existent, où K0 est l’amas contenant l’origine, nous montrons que

δd+(α1)d(α1)et2ηα1.

La borne inférieure sur δ est censée être précise pour d=1, α[13,1) et pour d=2, α[23,1], alors que la borne inférieure sur 2η est précise pour d=1, α(0,1), et pour α(0,1] pour d>1 : elle n’est probablement pas précise dans les autres cas. Notre outil principal est une relation entre les exposants critiques et l’isopérimétrie des cubes dans Zd.

Funding Statement

This work is supported by TopMath, the graduate program of the Elite Network of Bavaria and the graduate center of TUM Graduate School.

Acknowledgements

We thank an anonymous referee for useful comments.

Citation

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Johannes Bäumler. Noam Berger. "Isoperimetric lower bounds for critical exponents for long-range percolation." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 721 - 730, February 2024. https://doi.org/10.1214/22-AIHP1342

Information

Received: 20 July 2022; Revised: 31 October 2022; Accepted: 31 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718396
Digital Object Identifier: 10.1214/22-AIHP1342

Subjects:
Primary: 60K35 , 82B43
Secondary: 82B27

Keywords: Critical exponents , Long-range percolation , phase transition

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré

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Vol.60 • No. 1 • February 2024
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