Critical probability bounds for two-dimensional site percolation models

Abstract

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.

Upper and Lower Bounds for the Kagomé Lattice Bond Percolation Critical Probability

“…The largest solution of an upset probability equation in Table 3 is 0.529095, which corresponds to the upset consisting of all partitions except the minimal partition (Class 19). From Table 1, notice that K(π, 0.529095) > H(π) for partitions π in Classes 1, 2, 4, and 9, while K(π, 0.529095) < H(π) for partitions π in Classes 3,5,6,7,8,10,11,12,13,14,15,16,17,and 18. For each partition π in Classes 2, 4, and 9, we will construct a set R(π) consisting of π and specific partitions which are refinements of π.…”

“…Exact solutions have been found only for regular trees and a small number of periodic two-dimensional graphs [10,11,23,24]. For other graphs of interest, the problem has been approached by simulation and estimation, e.g., [19,20], and through rigorous bounds, e.g., [1,12,26,28]. A goal of these lines of research is to understand the dependence of the critical probability upon the detailed structure of the underlying graph and possibly to find accurate approximation formulae based on graph properties.…”

Bond percolation critical probability bounds for three Archimedean lattices

“…[21,22,30], and through rigorous bounds, e.g. [2,14,27,28]. A goal of these lines of research is to understand the dependence of the critical probability upon the detailed structure of the underlying graph and possibly to find accurate approximation formulae based on graph properties.…”

Interfaces determined by capillarity and gravity in a two-dimensional porous medium

“…that the dual event, finding an infinite cluster of open sites, has probability one, see e.g. [14], [10], [21]. In our setting, we have the following percolation result: …”