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

TL;DR

This paper investigates the upper transition point in percolation on the enhanced binary tree, providing a sharpened lower bound and discussing related solvable models, advancing understanding of hyperbolic structure percolation.

Contribution

It establishes a new lower bound for the upper transition point of the enhanced binary tree using renormalization-group methods and explores related solvable models.

Findings

Lower bound for $p_{c2}$ is approximately 0.55.

Discusses solvable models related to the enhanced binary tree.

Provides insights into percolation transitions on hyperbolic structures.

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.