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

Abstract

We perform Monte-Carlo simulations to study the Bernoulli ($p$) bond percolation on the enhanced binary tree which belongs to the class of nonamenable graphs with one end. Our numerical results show that the system has two different percolation thresholds $p_{c1}$ and $p_{c2}$. All the points in the intermediate phase $(p_{c1}<p<p_{c2})$ are critical and there exist infinitely many infinite clusters in the intermediate phase. In this phase the corresponding fractal exponent continuously increases with $p$ from zero to unity.

4 pages, 6 figures

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• Statistical mechanics of complex networks
• The structure and function of complex networks
• Critical phenomena in complex networks
• Percolation in the Hyperbolic Plane
• Phase Transitions on Nonamenable Graphs
• Geometric effects on critical behaviours of the Ising model
• Percolation on hyperbolic lattices
• Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane
• Ising model on hyperbolic lattice studied by corner transfer matrix renormalization group method
• The dynamic exponent of the Ising model on negatively curved surfaces

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• Phase Transition of the Ising model on a Hyperbolic Lattice
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
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• Hierarchical scale-free network is fragile against random failure
• Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
• Critical Phase of Bond Percolations on Growing Networks
• Analytic results for the percolation transitions of the enhanced binary tree
• Exact solution of the Nonconsensus Opinion Model on the line
• From explosive to infinite-order transitions on a hyperbolic network
• Ising model on a hyperbolic plane with a boundary
• Comment on `Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'
• Reply to Comment on `Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees'
• Robustness of correlated networks against propagating attacks
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network
• Bounds of percolation thresholds in the enhanced binary tree
• Upper transition point for percolation on the enhanced binary tree: A sharpened lower bound
• Phase transition without global ordering in a hierarchical scale-free network
• Bounds of percolation thresholds on hyperbolic lattices
• Teleportation Fidelity of Binary Tree Quantum Repeater Networks
• Random walks on bifractal networks