The Martin boundary of the Directed Landscape

TL;DR

This paper characterizes the Martin boundary of the directed landscape, showing it coincides with the horofunction boundary and describing the structure of eternal solutions and Busemann functions.

Contribution

It establishes the precise relationship between the Martin boundary, horofunction boundary, and Busemann functions in the directed landscape.

Findings

Martin boundary coincides with horofunction boundary

Minimal Martin boundary is given by Busemann functions

Every eternal solution is a max-plus convex combination of Busemann functions

Abstract

In the directed landscape, the Martin boundary coincides with the horofunction boundary. We show that functions in this boundary are precisely the eternal solutions possessing a spatial growth rate, and that the minimal Martin boundary is given by the Busemann functions. Moreover, every eternal solution can be expressed as a max-plus convex combination of countably many Busemann functions. Horofunctions are exactly those eternal solutions that admit a representation in terms of at most two Busemann functions with a common growth rate. As a consequence of instability, not all horofunctions are Busemann functions, and the Martin boundary is strictly larger than its minimal part.