A note on some critical thresholds of Bernoulli percolation

Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{c}\mathrm{u}\mathrm{t}}$ 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 $p_{\mathrm{c}\mathrm{u}\mathrm{t}}=p_c$ 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 ($\mathit{\text{Comm.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}\mathit{\text{Volume}}\mathit{\text{343,}}\mathit{\text{725-745.}}$ (2016)). We also study some applications for Bernoulli percolation on periodic trees.