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Consider Bernoulli bond percolation on a locally finite, connected graph $G$and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-momentmethod" 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 positiveanswer to Kahn's modified question. The key observation is that in Kahn'smodification, the new expectation quantity also appears in the differentialinequality of one-arm events. This links the question to a lemma ofDuminil-Copin and Tassion (\textit{Comm. Math. Phys. Volume 343, 725-745.}(2016)). We also study some applications for Bernoulli percolation on periodictrees.

A note on some critical thresholds of Bernoulli percolation