COMMENTS AND REPLIES: Comment on 'Monte Carlo simulation study of the twostage 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 (2009 J. Phys. A: Math. Theor. 42 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 November 2009
 DOI:
 10.1088/17518113/42/47/478001
 arXiv:
 arXiv:0910.4340
 Bibcode:
 2009JPhA...42U8001B
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 5 pages, 7 figures