Percolation thresholds in hyperbolic lattices.

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.