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 [1], 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 [6] 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. Let Z be the planar square lattice, i.e., the graph with vertex set Z in which each pair of nearest neighbours is joined by an edge. Let X = E(Z) be the edge-set of Z, and let Ω = {−1,+1}. We write ω = (ωe)e∈X for an element of Ω, and say that the edge e is open (in the state ω) if ωe = +1, and closed if ωe = −1. An event A ⊂ Ω is local if it depends on only finitely many coordinates. As usual, let Σ be the sigma-field generated by local events, and let Pp be the probability measure on (Ω,Σ) in which each edge is open with probability p, and these events are independent. Let θ(p) be the Pp-probability that the origin is in an infinite open cluster, i.e., an infinite connected subgraph C of Z with every edge of C open. In 1960, Harris [3] proved that θ(1/2) = 0; in 1980, Kesten [5] showed that θ(p) > 0 for p > 1/2, establishing that pc = 1/2 is the ‘critical probability’ for this model. A short proof of these results was given in [1], using a sharp-threshold result of Friedgut and Kalai [2], itself based on a result of Kahn, Kalai and Linial [4]. In 1982, Russo [6] proved a general sharp-threshold result (weaker than the more recent results described above) and applied it to percolation, to give a new proof of the ‘equality of critical probabilities’ for site percolation in Z. Although Russo does not explicitly say this, his application applies equally well to bond percolation, giving a new proof of the Harris-Kesten Theorem that seems not to be well known. Here we shall present Russo’s general sharp-threshold result, and then give a complete version of his application, to bond percolation in Z. Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA Trinity College, Cambridge CB2 1TQ, UK Research supported in part by NSF grant ITR 0225610 Royal Society Research Fellow, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK