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 = pc1, and there emerges a unique giant cluster at pc2 > pc1. 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 pc2 ≳ 0.55 by using phenomenological renormalization-group methods and discusses some solvable models related to the EBT.