We disprove a conjecture of Russ Lyons---that for every locally finite, connected graph $G$, the critical probability for (Bernoulli bond) percolation on $G$ is equal to the "first moment method" lower bound on this probability---and propose a possible alternative.
"Inequality of Two Critical Probabilities for Percolation." Electron. Commun. Probab. 8 184 - 187, 2003. https://doi.org/10.1214/ECP.v8-1099