Classification of half-planar maps

Abstract

We characterize all translation invariant half-planar maps satisfying a certain natural domain Markov property. For $p$-angulations with $p\ge3$ where all faces are simple, we show that these form a one-parameter family of measures $\mathbb{H}^{(p)}_{\alpha}$. 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 subcritical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.