The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds

Abstract

We describe how non-crossing partitions arise in substitution method calculations. By using efficient algorithms for computing non-crossing partitions we are able to substantially reduce the computational effort, which enables us to compute improved bounds on the percolation thresholds for three percolation models. For the Kagomé bond model we improve bounds from 0.5182 p c 0.5335 to 0.522197 p c 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 12 2 ) bond model we improve bounds from 0.7393 p c 0.7418 to 0.739773 p c 0.741125, improving the range from 0.0025 to 0.001352. We also improve the upper bound for the hexagonal site model, from 0.794717 to 0.743359.

Rigorous confidence intervals for critical probabilities

“…There are a small number of recent, more accurate, rigorous results, including an intervals of width 0.00135 and 0.0046 for site percolation on the (3,12 2 ) and Kagomé lattices obtained by May and Wierman [20].…”

A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

“…The substitution method compares probabilities of connections between vertices in the percolation model on an unsolved lattice with those on a solved reference lattice. Previous attempts to disprove the Tsallis conjecture, by May and Wierman [17,18], used the hexagonal lattice bond model as the reference model, since the hexagonal lattice has some structural similarity with the kagome lattice. Since then, a class of new exact solutions has been established.…”

“…We obtain mathematically rigorous bounds of 0.522394 < p c (kagome) < 0.526750, the electronic journal of combinatorics 22(2) (2015), #P2.52 which disprove the Tsallis conjecture by showing that in fact his value is strictly smaller than the kagome lattice bond percolation threshold. Note that the bounds are improvements of [18], and are approximately centered on the recent simulation estimates. Note also that, previous to the Tsallis conjecture, Wu [39] conjectured a percolation threshold of 0.524430 for the kagome lattice bond model.…”

“…Particularly notable is a mathematically rigorous 99.9999% confidence interval [0.52415, 0.52465] by Riordan and Walters [23]. Mathematically rigorous bounds for the percolation threshold have also improved over the years, as indicated in the following [18] This article is motivated by the disparity between the simulation estimates and the conjectured exact value by Tsallis [28], the root of p 3 − p 2 − p + 1 − 2 sin(π/18) = 0. For example, a recent estimate is 0.524404999134 by [9], while the numerical value of the Tsallis conjecture is 0.522372078 .…”

“…Our bounds are obtained by the substitution method, using stochastic ordering to compare an unsolved percolation model to an exactly-solved model. In addition to the kagome lattice bond model, it has been applied to several other percolation models [17,18,33,34,35,36]. The initial motivation for the development of the substitution method was to understand the approach of Ottavi, who proposed bounds of 0.522372 p c 0.528924 for the kagome lattice bond percolation threshold.…”