Percolation thresholds in hyperbolic lattices

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 .