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 $Q$.

Figure 1

Hyperbolic lattices $\{5,4\},\{4,5\}$ and $\{3,7\}$ (left to right).

Figure 2

Matching lattice $\widehat{G}$ of the $\{4,5\}$ hyperbolic lattice. The new edges, drawn as dashed lines, connect the vertices of each face in a clique.

Figure 3

Distribution of weights on site percolation invasion clusters of mass $N$ in the $\{3,7\}$ hyperbolic lattice.

Figure 4

$\frac{B\left(N\right)}{N}-\frac{B\left(2N\right)}{2N}\sim N^{-\delta }$ for bond percolation and $Q=7$.

Figure 5

Finite size scaling exponent $\delta$ for bond percolation. Note the convergence to its value $\delta =1$ for the tree.

Figure 6

The ratio $N/B\left(N\right)$ for invasion clusters of mass $N$ for site percolation on the $\{7,3\}$ hyperbolic lattice. The line is a fit of (9), and extrapolating to $N=\infty$ gives our estimate of $p_c^{\text{site}}$.

Figure 7

Scaling of $p_c^{\text{site}}\left(\{P,Q\}\right)$ with $P$.

Figure 8

Comparison of $p_c^{\text{site}}\left(\{P,Q\}\right)$ for $Q=7$ with the upper and lower bounds from Theorem t1.

Figure 9

Uniqueness threshold for $Q=7$ and $P=3,...,9$ and the bounds from Theorem t2.

Figure 10

Mean distance $\langle k_N\rangle$ of a vertex of the invasion cluster of mass $N$ to the origin. The lines $\{\infty ,Q\}$ represent the asymptotics (24) for the $Q$-regular tree. The data shown are for bond percolation, and the dotted lines are quadratic fits.

Figure 11

Scaling of the cluster density at criticality for the $\{7,3\}$ lattice using logarithmic binning. The line is $G_s\sim s^{1-\tau }$ for $\tau =5/2$, confirming that $\tau$ equals its mean-field value.

Figure 12

Types of vertices in layer $k$ and their edge labelings.

Figure 13

Types of vertices and their edge labelings for $P=3$.

Figure 14

Decay of the standard deviation $\sigma$ of $N/B\left(N\right)$ for $N=12800$ vs $P$.

Figure 15

Decay of the standard deviation $\sigma$ of $N/B\left(N\right)$ vs $N$.