Comment on `Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

TL;DR

This paper challenges previous claims about the second percolation threshold in enhanced binary trees, showing it is lower and does not follow the duality relation, with critical behavior similar to hyperbolic lattices.

Contribution

The study provides new Monte Carlo evidence that the second threshold in EBTs is lower than previously thought and does not obey the duality relation, aligning its critical behavior with hyperbolic lattices.

Findings

Second threshold p_{c2} is lower than previous estimates.

p_{c2} + p̄_{c1} < 1, violating duality.

Critical behavior matches that of hyperbolic lattices.

Abstract

The enhanced binary tree (EBT) is a nontransitive graph which has two percolation thresholds $p_{c1}$ and $p_{c2}$ with $p_{c1}<p_{c2}$. Our Monte Carlo study implies that the second threshold $p_{c2}$ is significantly lower than a recent claim by Nogawa and Hasegawa (J. Phys. A: Math. Theor. {\bf 42} (2009) 145001). This means that $p_{c2}$ for the EBT does not obey the duality relation for the thresholds of dual graphs $p_{c2}+\overline{p}_{c1}=1$ which is a property of a transitive, nonamenable, planar graph with one end. As in regular hyperbolic lattices, this relation instead becomes an inequality $p_{c2}+\overline{p}_{c1}<1$. We also find that the critical behavior is well described by the scaling form previously found for regular hyperbolic lattices.