Relations between site percolation thresholds

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.…”

“…Since U was arbitrary, 0.529095 is the largest solution solution for any upset, and thus is an upper bound for the Kagomé lattice bond model critical probability. H 19 (b) = 1 − 6b 4 − 6b 6 + 12b 7 + 3b 8 + 24b 9 − 18b 10 + 12b 11 − 71b 12 + 60b 13 − 93b 14 + 112b 15 + 69b 16 − 156b 17 + 57b 18…”

“…In constructing an optimal upset in the class poset, we prefer to include only classes of partitions which have higher probability in the Kagomé bond model than in the hexagonal lattice bond model at criticality. From Table 1, we see that only Classes 1,4,9,10,11,16,18, and 19 have greater Kagomé model probability for some p in [0.5182, 0.5209]. The upset in C which produces the smallest solution must be generated by a subset of these classes.…”

Critical probability bounds for two-dimensional site percolation models

“…A heuristic argument due to Shalitin (1982) suggests that if LA = L B the inequality holds, and that pc( L ) is near this upper bound.…”

An Improved Upper Bound for the Hexagonal Lattice Site Percolation Critical Probability

“…Stauffer [12] provided a Monte Carlo simulation estimate of 0.6962 and Shalitin [11] gave a heuristic method which derives a conjectured upper bound of 0.7072. This paper's upper bound of 0.79472 reduces the difference from the estimated value by nearly 12%, and the difference between the upper and lower bounds by more than 8%.…”