An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach. (September 2021)

• Record Type: Journal Article
• Authors: Yu, Gaoran; Wierman, John C.
• Is Part Of: Journal of applied probability. Volume 58:Number 3(2021)
• Page Start: 677
• Page End: 692
• Publication Date: 2021-09
• Subjects: Bond percolation -- percolation threshold -- cubic lattice -- body-centered cubic lattice -- projection; 60K35 -- 05C80 -- 82B43; 519.2
• DOI: 10.1017/jpr.2020.111
• Languages: English
• ISSNs: 0021-9002

Abstract

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.