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.

Percolation thresholds and Fisher exponents in hypercubic lattices

“…The limit (1) has been established rigorously for invasion bond percolation in Z 2 [10], but it is believed to hold on general lattices. Moreover, it appears to be an excellent numerical estimator for p c , and in [12] we used it to estimate p c to high precision on hyperbolic lattices. Based on these facts, we estimate p c by extrapolating the measured values N/B(N) to N = ∞.…”

“…We find that the convergence of this estimator is almost as good on d-dimensional lattices as it is on the tree. As in [12] we assume the form…”

Extremal $\{p, q\}$-Animals

“…However, the relevance of extremal animals in this setting to other extremal combinatorial problems has already been established. These include optimal disk packing problems [4], the calculation of Cheeger constants [9], the establishment of the exponential growth constant of regular hyperbolic tessellations [13], and the implementation of algorithms to sample certain hyperbolic animals to approximate hyperbolic percolation thresholds [22].…”

Topological percolation on hyperbolic simplicial complexes

“…Several works have studied percolation [26][27][28][44][45][46][47] and the Ising and Potts models [48,49] on other hierarchical networks, finding both continuous and discontinous phase transitions. However most of the results on percolation in hyperbolic networks [50,51] are restricted to d = 2 spaces. Here we explore how the scenario change in dimension d = 3 and we ad-dress the general question whether the nature of the transition changes with the dimension of the manifold.…”