We analyze the geometry of domain Markov half planar triangulations. In  it is shown thatthere exists a one-parameter family ofmeasures 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 inview of their geometry atα = 2/3. For α < 2/3, the maps form atree-like stricture with infinitely many small cut-sets.For α > 2/3,we obtain maps of hyperbolic nature with exponential growth andanchoredexpansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.
"Geometry and percolation on half planar triangulations." Electron. J. Probab. 19 1 - 28, 2014. https://doi.org/10.1214/EJP.v19-3238