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

@article{Baek2009CommentO,
  title={Comment on ‘Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees’},
  author={Seung Ki Baek and Petter Minnhagen and Beom Jun Kim},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2009},
  volume={42},
  pages={478001}
}
The enhanced binary tree (EBT) is a nontransitive graph which has two percolation thresholds pc1 and pc2 with pc1 < pc2. Our Monte Carlo study implies that the second threshold pc2 is significantly lower than a recent claim by Nogawa and Hasegawa (2009 J. Phys. A: Math. Theor. 42 145001). This means that pc2 for the EBT does not obey the duality relation for the thresholds of dual graphs which is a property of a transitive, nonamenable, planar graph with one end. As in regular hyperbolic… 
Reply to the comment on ‘Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees’
We discuss the nature of the two-stage percolation transition on the enhanced binary tree in order to explain the disagreement in the estimation of the second transition probability between the one
Bounds of percolation thresholds in the enhanced binary tree
Upper transition point for percolation on the enhanced binary tree: a sharpened lower bound.
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  • 2012
TLDR
This work presents the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree (EBT) as pc2 ≳ 0.55 by using phenomenological renormalization-group methods and discusses some solvable models related to the EBT.
Analytic results for the percolation transitions of the enhanced binary tree.
  • P. MinnhagenS. Baek
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
Percolation for a planar lattice has a single percolation threshold, whereas percolation for a negatively curved lattice displays two separate thresholds. The enhanced binary tree (EBT) can be viewed
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We investigate percolation and the Ising model on the hyperbolic plane through the means of Monte Carlo simulations and the Corner Transfer Matrix Renormalization Group (CTMRG), respectively.
Profile and scaling of the fractal exponent of percolations in complex networks
We propose a novel finite-size scaling analysis for percolation transition observed in complex networks. While it is known that cooperative systems in growing networks often undergo an infinite-order
Bounds of percolation thresholds on hyperbolic lattices.
  • Junghoon F LeeS. Baek
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
By applying the substitution method to known bounds of the order-5 pentagonal tiling, it is shown that p(c2) ≥ 0.382508 for theOrder-5 square tiler, p( c2) ≤ 0.472043 for its dual, and p(C2)≥ 0.275768 for the order -5-4 rhombille tiling.
Crossing on hyperbolic lattices.
  • H. GuR. Ziff
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
It is found that the crossing probability increases gradually from 0 to 1 as p increases from the lower p_{l} to the upper p_{u} critical values, and bounds and estimates are found for the values of p l and p u for these lattices.

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Reply to the comment on ‘Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees’
We discuss the nature of the two-stage percolation transition on the enhanced binary tree in order to explain the disagreement in the estimation of the second transition probability between the one
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