Constraint percolation on hyperbolic lattices

Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices, and they are interesting in their own right, with ordinary percolation exhibiting not one but two phase transitions. We study four constraint percolation models—$k$-core percolation (for $k=1,2,3$) and force-balance percolation—on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggest that all of the $k$-core models, even for $k=3$, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the $k$-core percolation models.

Figure 1

$\{3,7\}$ tessellation on the Poincaré disk.

Figure 2

$\{3,7\}$ tessellation on the Poincaré disk with the four boundary regions.

Figure 3

One cannot embed a tree of connectivity $z=6$ on the $\{3,8\}$ tessellation due to the lack connections.

Figure 4

Tessellation $\{4,8\}$ enables the construction of trees of connectivity $z=6$.

Figure 5

Tree construction on tessellations $\{4,7\}$ and $\{4,8\}$: Top: tree of connectivity $z=5$ on the $\{4,7\}$ tessellation. Bottom: tree of connectivity $z=6$ on the $\{4,8\}$ tessellation.

Figure 6

Euclidean illustration of the central part of the trees on hyperbolic tessellations: Left: Euclidean illustration of the “central” part of the $z=5$ tree on tessellation $\{4,7\}$. Right: Euclidean illustration of the “central” part of the $z=6$ tree on tessellation $\{4,8\}$.

Figure 7

Illustration of all the possible cases of occupation for a $k=4$-core cluster on a tree of connectivity $z=5$.

Figure 8

Crossing probability on $\{3,7\}$ tessellation for the different percolation models: (a) $k=1$ core, (b) $k=2$ core, (c) $k=3$ core, and (d) force balance.

Figure 9

Inverse slope of the crossing probability at the inflection point, $1/M_0$, as a function of $1/l$ for the different percolation models on the $\{3,7\}$ lattice. For the 1-core model, $1/M_0$ tends to $0.240\pm 0.004$; for 2-core, to $0.223\pm 0.010$; for 3-core, to $0.131\pm 0.007$; and for force-balance (FB), to $-0.016\pm 0.02$, indicating $M_0$ is tending to $\infty$ as $l$ tends to $\infty$.

Figure 10

The fractional size of the largest cluster $P_{\text{lc}}$ for the different percolation models on the $\{3,7\}$ lattice: (a) $k=1$ core, (b) $k=2$ core, (c) $k=3$ core, and (d) force balance.

Figure 11

The inverse of the maximum slope of $P_{\infty }$ as a function of $\ell$ for the different models' percolation on the $\{3,7\}$ lattice.

Figure 12

Inverse slope of $P_{\infty }, 1/S$, tendency on $1/l$ for points on the line $P_{\infty }=c$ and $c=0.3,0.4,0.5,0.6,0.7$, for the force-balance model on the $\{3,7\}$ lattice.

Figure 13

Culling time for the different constraint percolation models for the $\{3,7\}$ lattice: (a) $k=2$ core, (b) $k=3$ core, and (c) force balance. Each data set was averaged over 50 000 samples.

Figure 14

Behavior of the width $\sigma$ vs $1/l$ for the 2-core, 3-core, and force balance models on the $\{3,7\}$ lattice.

Figure 15

Ratio $S_2/S_1$ for $k=1$ core and for the tessellation $\{3,7\}$.

Figure 16

Cluster size distribution $n_s$ for the different percolation models in the $\{3,7\}$ lattice: (a) $k=1$ core, (b) $k=2$ core, (c) $k=3$ core, and (d) force balance.