The Martin boundary of the Directed Landscape

Firas Rassoul-Agha; Mikhail Sweeney

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arXiv · 2605.21366

The Martin boundary of the Directed Landscape

Firas Rassoul-Agha·Mikhail Sweeney

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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.

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BibTeX RIS

Firas Rassoul-Agha, Mikhail Sweeney. 2026-05-20. The Martin boundary of the Directed Landscape. https://arxiv.org/abs/2605.21366

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