exact relation to the thresholds of its dual lattice.
author:
- Petter Minnhagen
- Seung Ki Baek
title: Analytic results for the percolation transitions of the enhanced binary tree
---

Introduction
============

The percolation transition is one of the main examples of a critical phenomena [@stauffer; @chris], and there have been extensive studies on lattices in Euclidean space, in particular, various types of planar lattices. An interesting and somewhat less studied aspect of the percolation phenomena is that, while a planar lattice possesses a single percolation threshold, a hyperbolic lattice with a constant negative curvature has two different thresholds [@baek]. Figure \[fig:poin\](a) gives an example of a hyperbolic lattice with Schläfli symbol $\{7,3\}$ as well as its dual, $\{3,7\}$, where $\{m,n\}$ means that the number of regular $m$-gons meeting at each vertex inside the structure is $n$. One characteristic feature of negative curved lattices is that the number of boundary points is a finite fraction of the total number of lattice points. The fact that the number of surface points is a finite fraction is intimately connected to the existence of two thresholds: the probability of reaching *one* of the boundary point from the midpoint becomes finite at occupation probability 