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.

5 pages, 7 figures

References in corpus (2)

• Percolation on hyperbolic lattices
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees

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• Analytic results for the percolation transitions of the enhanced binary tree
• Reply to Comment on `Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees'
• 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