Critical Percolation Probabilities

Abstract

This paper summarizes the current state of knowledge of critical probabilities in percolation models. There are four common definitions of critical probability in the literature, which are known to be unequal for some graphs. A heuristic method of Sykes and Essam [17] has produced correct critical probability values for a few planar lattice models. However, counterexamples have been constructed to some conclusions of the Sykes and Essam method, and the method is valid only for two-dimensional graphs. Exact critical probabilities have been rigorously determined for a few two-dimensional percolation models, using techniques of Seymour and Welsh [15], Russo [14], and Kesten [10] involving crossing probabilities of rectangles.