Geometry and percolation on half planar triangulations

TL;DR

This paper investigates the geometric properties of half planar triangulations, revealing a phase transition at a critical parameter and analyzing the implications for percolation clusters and random walks.

Contribution

It characterizes the geometric phase transition of domain Markov half planar triangulations and explores their percolation and random walk behaviors.

Findings

Maps are tree-like for <2/3 with small cut-sets

Maps are hyperbolic for \u03b1>2/3 with exponential growth

Results on percolation clusters and random walks on these maps

Abstract

We analyze the geometry of domain Markov half planar triangulations. In \cite{AR13} it is shown that there exists a one-parameter family of measures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition in view of their geometry at $\alpha = 2/3$. For $\alpha<2/3$, the maps form a tree-like stricture with infinitely many small cut-sets. For $\alpha > 2/3$, we obtain maps of hyperbolic nature with exponential growth and anchored expansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.