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

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.