Analytic results for the percolation transitions of the enhanced binary tree

Percolation for a planar lattice has a single percolation threshold, whereas percolation for a negatively curved lattice displays two separate thresholds. The enhanced binary tree (EBT) can be viewed as a prototype model displaying two separate percolation thresholds. We present an analytic result for the EBT model which gives two critical percolation threshold probabilities, $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}$ and $p_{c2}=1/2$, and yields a size-scaling exponent $\Phi =\text{ln}\left[\frac{p\left(1+p\right)}{1-p\left(1-p\right)}\right]/\text{ln}2$. It is inferred that the two threshold values give exact upper limits and that $p_{c1}$ is furthermore exact. In addition, we argue that $p_{c2}$ is also exact. The physics of the model and the results are described within the midpoint-percolation concept: Monte Carlo simulations are presented for the number of boundary points which are reached from the midpoint, and the results are compared to the number of routes from the midpoint to the boundary given by the analytic solution. These comparisons provide a more precise physical picture of what happens at the transitions. Finally, the results are compared to related works, in particular, the Cayley tree and Monte Carlo results for hyperbolic lattices as well as earlier results for the EBT model. It disproves a conjecture that the EBT has an exact relation to the thresholds of its dual lattice.

Figure 1

Hyperbolic lattice structures drawn on the Poincaré disk. (a) The solid lines show a heptagonal lattice, {7,3}, and the dotted lines show its dual, {3,7}. (b) A tree structure, represented as $\left\{\infty ,3\right\}$, up to the maximum level $L=10$. (c) The enhanced tree obtained from (b) by connecting the tree branches.

Figure 2

Schematic representations of (a) the binary tree and (b) EBT. Arrows show examples of routes from the surface to the midpoint. In panel (a), there is only one possible route from a surface point to the top, whereas in panel (b), there are several because the tree branches are connected. (c) A horizontal layer with a rightward tagged route. The three black dots denote the first point of the route on level $l$, the last point of route on level $l$, and the first point of the route on level $l-1$, respectively.

Figure 3

(Color online) The minimum number of backsteps $n_b$ for a nonzero-backstep path which connects from bottom to the top. (a) The data is plotted as $n_b/L$ against $L$ in a log-log plot for a sequence of fixed $p$ values. For smaller $p$, the data shows that $n_b/L$ vanishes as a power law $n_b/L\propto L^{-\alpha }$ with increasing $L$. However, for $p\approx 0.5$, it instead goes to a constant. This change of behavior signals a phase transition. (b) The numerically determined power-law index $\alpha$ plotted as a function of $p$. The dotted line presents a second-order polynomial fit for eyes guidance. Extrapolation for the data points suggests that $\alpha$ vanishes at $p=0.50\left(1\right)$.

Figure 4

(Color online) Comparisons between $r^{\text{tag}}$, the average number of tagged routes from an arbitrary surface point to the midpoint, and, $b$, the number of surface points connected to the midpoint. $r^{\text{tag}}$ is given by the analytical solution and $b$ is obtained from numerical simulations. (a) The number of tagged routes from the surface to the midpoint, $B{r}^{\text{tag}}$ for various sizes $L$. All curves cross in a single point at $p_{c1}\approx 0.303$. (b) The same plot for $b$. Note the strong resemblance and the single crossing point at $p\approx 0.303\left(1\right)$. The panel (c) illustrates the close connection between $B{r}^{\text{tag}}$ and $b$. For $p\le 0.4$ the number of routes and the number of connections are almost the same (the number of routes is slightly smaller as explained in the text). However for $p\ge 0.4$, the number of routes exceeds the number of connected surface points. The panel (d) illustrates this connection further by comparing the size-scaling exponent $1+\Phi$ obtained analytically for $r^{\text{tag}}$ with the size scaling for $b$ obtained from (b). The agreement is striking for $p\le 0.4$. (e) $r^{\text{tag}}$ for various sizes with the crossing point at $p_{c2}=1/2$. (f) The corresponding crossing point for $b/B$ obtained numerically. The crossing point is close to 0.50, but the size-scaling exponent $\Phi$ is different. For $r^{\text{tag}}$ the value is $\Phi \left(p_{c2}\right)=0$ but for $b$ it is ${\Phi }^{\prime }\left(p_{c2}\right)\approx -0.11$, as explained in the text.