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arXiv · 1212.4916

Bounds of percolation thresholds on hyperbolic lattices

Junghoon F. Lee·Seung Ki Baek
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Abstract

We analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative Gaussian curvature. The quantity of our main concern is $p_{c2}$, the value of occupation probability where a unique unbounded cluster begins to emerge. By applying the substitution method to known bounds of the order-5 pentagonal tiling, we show that $p_{c2} \ge 0.382 508$ for the order-5 square tiling, $p_{c2} \ge 0.472 043$ for its dual, and $p_{c2} \ge 0.275 768$ for the order-5-4 rhombille tiling.

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Junghoon F. Lee, Seung Ki Baek. 2012-12-28. Bounds of percolation thresholds on hyperbolic lattices. https://doi.org/10.1103/physreve.86.062105

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