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An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach

Part of: Graph theory Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  16 September 2021

Gaoran Yu  and
John C. Wierman
Gaoran Yu*
Affiliation:
Johns Hopkins University
John C. Wierman*
Affiliation:
Johns Hopkins University
*
*Postal address: Department of Applied Mathematics and Statistics, Wyman Park Building, Fourth Floor, Johns Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA.
*Postal address: Department of Applied Mathematics and Statistics, Wyman Park Building, Fourth Floor, Johns Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA.
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Abstract

We reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.

Keywords

Bond percolationpercolation thresholdcubic latticebody-centered cubic latticeprojection

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs 82B43: Percolation
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 677 - 692
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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