Existence, uniqueness and coalescence of directed planar geodesics: proof via the increment-stationary growth process

Abstract

We present a proof of the almost sure existence, uniqueness and coalescence of directed semi-infinite geodesics in planar growth models that is based on properties of an increment-stationary version of the growth process. The argument is developed in the context of the exponential corner growth model. It uses coupling, planar monotonicity, and properties of the stationary growth process to derive the existence of Busemann functions, which in turn control geodesics. This soft approach is in some situations an alternative to the much-applied 20-year-old arguments of C. Newman and co-authors. Along the way we derive some related results such as the distributional equality of the directed geodesic tree and its dual, originally due to L. Pimentel.