Classification of Half Planar Maps

TL;DR

This paper characterizes all translation invariant half planar maps with a domain Markov property, identifies a family of measures for p-angulations, and reveals a phase transition in triangulations affecting their properties.

Contribution

It introduces a comprehensive classification of half planar maps satisfying specific invariance and Markov properties, including phase transition phenomena for triangulations.

Findings

Family of measures H^{(p)}_{alpha} for p-angulations

Existence of phase transition in triangulations

Critical maps are the half plane uniform infinite planar maps

Abstract

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.