A note on the Harris–Kesten Theorem

A short proof of the Harris–Kesten result that the critical probability for bond percolation in the planar square lattice is 1/2 was given in [B. Bollobás, O.M. Riordan, A short proof of the Harris–Kesten Theorem, Bull. London Math. Soc. 38 (2006) 470–484], using a sharp-threshold result of Friedgut and Kalai. Here we point out that a key part of this proof may be replaced by an argument of Russo [L. Russo, An approximate zero–one law, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 61 (1982) 129–139] from 1982, using his approximate zero–one law in place of the Friedgut–Kalai result. Russo’s paper gave a new proof of the Harris–Kesten Theorem that seems to have received little attention.