Analytic results for the percolation transitions of the enhanced binary tree

TL;DR

This paper analytically determines the two percolation thresholds of the enhanced binary tree, providing exact values and a size-scaling exponent, and compares these results with simulations and related models to deepen understanding of percolation on curved structures.

Contribution

It presents the first analytic derivation of the two critical percolation thresholds for the enhanced binary tree, clarifying their exactness and physical implications.

Findings

Derived exact thresholds: p_{c1}=1/2√13 - 3/2 and p_{c2}=1/2.

Established the size-scaling exponent Φ.

Validated the analytic results with Monte Carlo simulations.

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.