Analytic results for the percolation transitions of the enhanced binary tree

Abstract

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, $p_{c1}=1/2\sqrt{13}-3/2$ and $p_{c2}=1/2$, and yields a size-scaling exponent $\Phi =\ln [\frac{p(1+p)}{1-p(1-p)}]/\ln 2$. 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.

12 pages, 14 figures

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• From explosive to infinite-order transitions on a hyperbolic network
• Bounds of percolation thresholds in the enhanced binary tree
• Upper transition point for percolation on the enhanced binary tree: A sharpened lower bound
• Bounds of percolation thresholds on hyperbolic lattices
• Hierarchical renormalization-group study on the planar bond-percolation problem
• Teleportation Fidelity of Binary Tree Quantum Repeater Networks