Open Access
2023 A note on some critical thresholds of Bernoulli percolation
Pengfei Tang
Author Affiliations +
Electron. J. Probab. 28: 1-22 (2023). DOI: 10.1214/23-EJP926


Consider Bernoulli bond percolation on a locally finite, connected graph G and let pcut be the threshold corresponding to a “first-moment method” lower bound. Kahn (Electron. Comm. Probab. Volume 8, 184-187. (2003)) constructed a counter-example to Lyons’ conjecture of pcut=pc and proposed a modification. Here we give a positive answer to Kahn’s modified question. The key observation is that in Kahn’s modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (Comm. Math. Phys. Volume 343, 725-745. (2016)). We also study some applications for Bernoulli percolation on periodic trees.

Funding Statement

Supported by the National Science Foundation under grant DMS-1612363, DMS-1954086 and by ERC starting grant 676970 RANDGEOM.


We thank Russ Lyons for many helpful discussions and support. Most of the work was done while the author was a graduate student at Indiana University. We also thank Wai-Kit Yeung for discussions and the reference on semi-algebraic functions. We are also grateful to the anonymous referee for his/her careful reading and helpful comments which improve the presentation a lot.


Download Citation

Pengfei Tang. "A note on some critical thresholds of Bernoulli percolation." Electron. J. Probab. 28 1 - 22, 2023.


Received: 21 May 2022; Accepted: 1 March 2023; Published: 2023
First available in Project Euclid: 2 March 2023

MathSciNet: MR4555229
zbMATH: 07707070
Digital Object Identifier: 10.1214/23-EJP926

Primary: 60CO5 , 60K35

Keywords: Bernoulli percolation , critical probability , cutset , periodic trees

Vol.28 • 2023
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