Generalized cell-dual-cell transformation and exact thresholds for percolation.
Suggested by Scullard's recent star-triangle relation for correlated bond systems, we propose a general "cell-dual-cell" transformation, which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated. We directly verify Scullard's site percolation thresholds, and derive the bond thresholds for his "martini" lattice (pc = 1/square root 2) and the "A" lattice (pc = 0.625457..., solution to p5 - 4p4 + 3p3 + 2p2 - 1 = 0). We also present a precise Monte Carlo test of the site threshold for the "A" lattice.
Ziff
Robert M
RM
Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA.