  We characterize all translation invariant half planar maps satisfying a
certain natural domain Markov property. For p-angulations with p \ge 3 where
all faces are simple, we show that these form a one-parameter family of
measures H^{(p)}_{alpha}. For triangulations we also establish existence of a
phase transition which affects many properties of these maps. The critical maps
are the well known half plane uniform infinite planar maps. The sub-critical
maps are identified as all possible limits of uniform measures on finite maps
with given boundary and area.
