Bounds of percolation thresholds on hyperbolic lattices

TL;DR

This paper analytically derives bounds for the percolation threshold where a unique infinite cluster appears on various hyperbolic lattices, advancing understanding of percolation in negatively curved spaces.

Contribution

It provides new lower bounds for the critical occupation probability $p_{c2}$ on several hyperbolic lattices using the substitution method.

Findings

Lower bounds for $p_{c2}$ on order-5 square tiling and its dual.

Lower bound for $p_{c2}$ on order-5-4 rhombille tiling.

Application of substitution method to hyperbolic lattice percolation.

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.