  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.
