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Generalized cell/dual-cell transformation for percolation, and new exact thresholds

Robert M. Ziff
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Abstract

Suggested by Scullard's recent star-triangle relation for bond correlated 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 new site percolation thresholds, and derive the bond thresholds for his "martini" lattice (pc = 1/sqrt(2)) and the "A" lattice (pc = 0.625457..., solution to p^5 - 4 p^4 + 3 p^3 + 2 p^2 - 1 = 0). We also present a precise Monte-Carlo test the site threshold for the "A" lattice.

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Robert M. Ziff. 2005-12-12. Generalized cell/dual-cell transformation for percolation, and new exact thresholds. https://arxiv.org/abs/cond-mat/0510245

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