Geometry and percolation on half planar triangulations

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 $α= 2/3$. For $α<2/3$, the maps form a tree-like stricture with infinitely many small cut-sets. For $α> 2/3$, 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.

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