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doi:10.1088/1751-8113/42/47/478001

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

Seung Ki Baek et al 2009 J. Phys. A: Math. Theor. 42 478001 (4pp) doi: 10.1088/1751-8113/42/47/478001

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Seung Ki Baek¹, Petter Minnhagen¹ and Beom Jun Kim²
¹ Department of Theoretical Physics, Umeå University, 901 87 Umeå, Sweden
² BK21 Physics Research Division and Department of Energy Science, Sungkyunkwan University, Suwon 440-746, Korea
E-mail: beomjun@skku.edu

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.

PACS numbers: 64.60.ah, 02.40.Ky, 05.70.Fh

Print publication: Issue 47 (27 November 2009)
Received 8 May 2009
Published 4 November 2009

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

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