2005/07/31 by Christian R. Scullard, Chris Scullard · 4 citations
Chemistry · Mathematics · Physics and Astronomy · #Chemistry #Combinatorics #Condensed matter physics #Crystallography #Hexagonal crystal system #Hexagonal lattice #Integer lattice #Lattice (music) #Mathematics #Percolation threshold #Physics #Quantum mechanics #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn
paper · pdf · doi:10.1103/physreve.73.016107
12 pages, 10 figures. Submitted to Physical Review E
arxiv created 2005/08/09 · openalex publication_date 2006/01/10 · arxiv updated 2013/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
I construct a two-dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined. The primary lattice studied here, the "martini lattice," is a hexagonal lattice with every second site transformed into a triangle. The site threshold of this lattice is found to be 0.764826..., i.e., the solution to p4 - 3p3 + 1 = 0, while the others have (square root 5 - 1)/2 (the inverse of the golden ratio) and 1/square root 2. This last solution suggests a possible approach to establishing the bound for the hexagonal site threshold, pc < 1/square root 2. To derive these results, I solve a correlated bond problem on the hexagonal lattice by use of the star-triangle transformation and then, by a particular choice of correlations derived from a site-to-bond transformation, solve the site problem on the martini lattice.