Consider Bernoulli bond percolation on a locally finite, connected graph G and let 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 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.
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.
"A note on some critical thresholds of Bernoulli percolation." Electron. J. Probab. 28 1 - 22, 2023. https://doi.org/10.1214/23-EJP926