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PreprintarXiv licencearXiv

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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Cite

BibTeXRIS
BibTeX
@misc{rassoulagha2026martin,
  title = {{The Martin boundary of the Directed Landscape}},
  author = {Firas Rassoul-Agha and Mikhail Sweeney},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.21366},
  eprint = {2605.21366},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.21366},
}
RIS
TY  - UNPB
TI  - The Martin boundary of the Directed Landscape
AU  - Firas Rassoul-Agha
AU  - Mikhail Sweeney
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2605.21366
UR  - https://arxiv.org/abs/2605.21366
AB  - 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.
ER  -