A note on the Harris-Kesten Theorem

  • Authors:

  • Béla Bollobás;Oliver Riordan

  • Affiliations:

  • Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA and Trinity College, Cambridge CB2 1TQ, UK;Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

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. Bollobas, 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.