Upper transition point for percolation on the enhanced binary tree: A sharpened lower bound

Abstract

Hyperbolic structures are obtained by tiling a hyperbolic surface with negative Gaussian curvature. These structures generally exhibit two percolation transitions: a system-wide connection can be established at a certain occupation probability $p=p_{c1}$ and there emerges a unique giant cluster at $p_{c2}>p_{c1}$. There have been debates about locating the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree (EBT), which is constructed by adding loops to a binary tree. This work presents its lower bound as $p_{c2}\gtrsim 0.55$ by using phenomenological renormalization-group methods and discusses some solvable models related to the EBT.

12 pages, 20 figures

References in corpus (11)

• Critical Percolation in Finite Geometries
• Small-World Bonds and Patchy Percolation on the Hanoi Network
• Percolation on hyperbolic lattices
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks
• Analytic results for the percolation transitions of the enhanced binary tree
• 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'
• Bounds of percolation thresholds in the enhanced binary tree
• Conformal Field Theory and Hyperbolic Geometry
• Hierarchical renormalization-group study on the planar bond-percolation problem

Cited by in corpus (4)

• Crossing on hyperbolic lattices
• Profile and scaling of the fractal exponent of percolations in complex networks
• Bounds of percolation thresholds on hyperbolic lattices
• Constraint percolation on hyperbolic lattices