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

Authors:
William D. May;John C. Wierman
Affiliations:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore MD, USA (e-mail: wdmay@jhu.edu, wierman@jhu.edu);Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore MD, USA (e-mail: wdmay@jhu.edu, wierman@jhu.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2007

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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 $0.5182 \leq p_c \leq 0.5335$ to $0.522197 \leq p_c \leq 0.526873$, improving the range from 0.0153 to 0.004676. For the $(3,12^2)$ bond model we improve bounds from $0.7393 \leq p_c \leq 0.7418$ to $0.739773 \leq p_c \leq 0.741125$, 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.