Critical probability bounds for two-dimensional site percolation models

Abstract. We present three techniques for determining rigorous bounds for site percolation critical probabilities of two-dimensional lattices. A technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that p, ( D) L 0.5020 for the dice lattice D. Combining this method with Kesten's duality result simplifies Toth's derivation of the lower bound p, ( S) 2 0.5034 for the square lattice S. We also present a technique for deriving upper bounds for bipartite graphs. A technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice H: p, ( H) G 0.8079 and p, ( H) G. i p, ( S). The grouping technique is applied to the dice lattice to find the upper bound p, ( D) G 0.7937. 1-1