1212.4916 (Junghoon F. Lee et al.)

Bounds of percolation thresholds on hyperbolic lattices    [PDF]

Junghoon F. Lee, Seung Ki Baek

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.

View original: http://arxiv.org/abs/1212.4916