Sharp thresholds and percolation in the plane

Recently, it was shown by Bollobás and Riordan [Probab Theory Related Fields 136 (2006), 417-468] that the critical probability for random Voronoi percolation in the plane is 1/2. As a by-product of the method, a short proof of the Harris-Kesten Theorem was given by Bollobás and Riordan [Bull London Math Soc 38 (2006), 470-484]. The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results and to prove new ones. © 2006 Wiley Periodicals, Inc