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Combinatorics, Probability and Computing Combinatorics, Probability and Computing

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

Published online by Cambridge University Press:  01 March 2007

WILLIAM D. MAY  and
JOHN C. WIERMAN
WILLIAM D. MAY
Affiliation:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore MD, USA (e-mail: wdmay@jhu.edu, wierman@jhu.edu)
JOHN C. WIERMAN
Affiliation:
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore MD, USA (e-mail: wdmay@jhu.edu, wierman@jhu.edu)
Corresponding
E-mail address:
wdmay@jhu.edu
wierman@jhu.edu
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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 ≤ pc ≤ 0.5335 to 0.522197 ≤ pc ≤ 0.526873, improving the range from 0.0153 to 0.004676. For the (3, 122) bond model we improve bounds from 0.7393 ≤ pc ≤ 0.7418 to 0.739773 ≤ pc ≤ 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.

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Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 285 - 307
Copyright
Copyright © Cambridge University Press 2006

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