Percolation on hyperbolic lattices

The percolation transitions on hyperbolic lattices are investigated numerically using finite-size scaling methods. The existence of two distinct percolation thresholds is verified. At the lower threshold, an unbounded cluster appears and reaches from the middle to the boundary. This transition is of the same type and has the same finite-size scaling properties as the corresponding transition for the Cayley tree. At the upper threshold, on the other hand, a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections. The finite-size scaling properties for this upper threshold are different from those of the Cayley tree and two of the critical exponents are obtained. The results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own.

Figure 1

(Color online) Scaling plots for regular square lattices, using (a) the ratio between the largest and the second largest cluster sizes, and (b) the number of boundary points connected to the midpoint of the lattice. The two-dimensional percolation scaling exponent $\nu =4∕3$ and $\kappa =\nu ∕(1+\nu )=4∕7$ are used for scaling collapse with $p_u=p_c=1∕2$. The average is taken over ${10}^6$ independent realizations for each plot.

Figure 2

(Color online) Poincaré disk representation of the hyperbolic lattice $\{m,n\}=\{3,7\}$ [seven $(n=7)$ triangles $(m=3)$ meet at every vertex] with the total number of layers $l=6$. Solid lines show occupied bonds in a realization with the occupation probability of $p=0.2$.

Figure 3

(Color online) Numerical results in Cayley trees, averaged over ${10}^6$ realizations. (a) The ratio between $s_2$ and $s_1$. Inset: A closer look near $p=1$, where one can hardly see any size dependency. (b) Scaling plot by Eq. (2). The scaling function is well-described by $\frac{3}{2}\exp \left[2l(p-p_m)\right]$ near $p_m=0.50$. (c) $\varphi$ as a function of $p$, provided that $b\left(p\right)\sim B^{\varphi }$. The horizontal axis is in the log scale, confirming $\varphi =1+{\log }_{2}p$. (d) Cluster size distribution at various $p$ with $l=15$. It approaches to $P\left(s\right)\sim s^{-2}$ as $p\to 1$, while finiteness of the system adds an exponential cutoff.

Figure 4

(Color online) Numerical results for heptagonal lattices, {7, 3}, averaged over ${10}^6$ realizations. (a) Measurement of $b$ yields a consistent result with the tree case. (b) The ratio between two largest cluster sizes $s_2∕s_1$ gives $p_u\approx 0.72$, and (c) extrapolating $b∕B$ gives $p_b\approx 0.72$, supporting $p_b=p_u$. (d) The exponent $\varphi$ as a function of $p$ shows a clear difference from the tree case [see Fig. 3c, for comparison]. At $p\gtrsim 0.84$, $\varphi$ largely fluctuates as it is hard to determine the size dependency from the numerical data.

Figure 5

(Color online) Scaling plot of $b∕B$ in {7,3} using Eq. (4), with $p_b=0.72$, $\varphi =0.82$, and $1∕\overline{\nu }=0.12$.

Figure 6

(Color online) (a) Cluster size distribution, $P\left(s\right)$, at various $p$ in {7,3}. It shows a power-law form also above $p_u$. (b) A schematic view of the Poincaré disk centered at $O$. The lattice is constructed up to the dashed circle, which is apart from the circumference of the disk by $\overline{AB}=\overline{CD}=\delta$ in the Euclidean measure.

Figure 7

(Color online) Numerical results for {4,5}, averaged over ${10}^6$ realizations. (a) Scaling plot with $p_m=0.27$. (b) Scaling collapse with $p_b=0.52$, $\varphi =0.82$, and $1∕\overline{\nu }=0.11$.

Figure 8

(Color online) Numerical results for {3,7}, averaged over ${10}^6$ realizations. Here are shown the scaling plots (a) at $p_m=0.20$ and (b) at $p_b=0.37$ with $\varphi =0.82$ and $1∕\overline{\nu }=0.11$.