Percolation Thresholds in Hyperbolic Lattices

TL;DR

This paper accurately computes percolation thresholds in hyperbolic lattices using invasion percolation, explores their dependency on lattice parameters, and establishes bounds and critical exponents.

Contribution

It provides highly precise numerical values for percolation thresholds in hyperbolic tessellations and derives bounds and scaling relations based on lattice parameters.

Findings

Percolation thresholds are computed with six to seven decimal place accuracy.

Functional dependency of thresholds on P and Q is numerically explored.

Rigorous bounds for thresholds are established, aiding in understanding their scaling.

Abstract

We use invasion percolation to compute numerical values for bond and site percolation thresholds $p_c$ (existence of an infinite cluster) and $p_u$ (uniqueness of the infinite cluster) of tesselations $\{P,Q\}$ of the hyperbolic plane, where $Q$ faces meet at each vertex and each face is a $P$-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on $P$ and $Q$ and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for $p_c$ and $p_u$ that can be used to find the scaling of both thresholds as a function of $P$ and $Q$.