A note on some critical thresholds of Bernoulli percolation

TL;DR

This paper investigates critical thresholds in Bernoulli percolation, confirming a modified conjecture by Kahn and connecting it to recent mathematical lemmas, with applications to periodic trees.

Contribution

It provides a positive answer to Kahn's modified question on percolation thresholds and links the expectation-based bounds to differential inequalities, advancing understanding of percolation phase transitions.

Findings

Confirmed Kahn's modified conjecture on percolation thresholds.

Linked expectation quantities to differential inequalities of one-arm events.

Applied results to Bernoulli percolation on periodic trees.

Abstract

Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons' conjecture of $p_{\mathrm{cut}}=p_c$ 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 (\textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees.