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arxiv: 2605.21366 · v1 · pith:HC7RL444new · submitted 2026-05-20 · 🧮 math.PR · math.DS· math.MG

The Martin boundary of the Directed Landscape

Firas Rassoul-Agha , Mikhail Sweeney This is my paper

Pith reviewed 2026-05-21 03:21 UTC · model grok-4.3

classification 🧮 math.PR math.DSmath.MG
keywords directed landscapeMartin boundaryhorofunction boundaryBusemann functionseternal solutionsmax-plus convex combinationinstability property
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The pith

In the directed landscape the Martin boundary coincides with the horofunction boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that in the directed landscape the Martin boundary is identical to the horofunction boundary. Boundary functions turn out to be exactly the eternal solutions that carry a spatial growth rate. The minimal part of the boundary consists of Busemann functions, while every eternal solution decomposes as a max-plus convex combination of countably many such functions. Horofunctions themselves reduce to representations using at most two Busemann functions that share one growth rate. The instability property then forces the full boundary to be strictly larger than this minimal collection.

Core claim

The Martin boundary of the directed landscape coincides with the horofunction boundary. Functions in this boundary are precisely the eternal solutions possessing a spatial growth rate, and 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.

What carries the argument

The Martin boundary identified with the horofunction boundary through eternal solutions that possess spatial growth rates and are built from Busemann functions by max-plus convex combinations.

If this is right

  • Every eternal solution decomposes into a countable max-plus combination of Busemann functions.
  • Horofunctions admit representations using at most two Busemann functions of common growth rate.
  • The Martin boundary properly contains its minimal part given by the Busemann functions.
  • Instability implies that not every horofunction is itself a Busemann function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit max-plus decomposition may let researchers build new eternal solutions in related growth models by superposing known Busemann functions.
  • Methods developed for Martin boundaries in other settings can now be transferred directly to the horofunction side of the directed landscape.
  • The countable combination structure suggests possible numerical schemes that approximate general eternal solutions from finite collections of Busemann functions.

Load-bearing premise

The directed landscape admits eternal solutions possessing a spatial growth rate together with an instability property that prevents every horofunction from being a Busemann function.

What would settle it

An explicit eternal solution in the directed landscape that cannot be written as a max-plus convex combination of countably many Busemann functions would refute the decomposition statement.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that in the directed landscape the Martin boundary coincides with the horofunction boundary. Functions in this boundary are precisely the eternal solutions possessing a spatial growth rate; the minimal Martin boundary consists of the Busemann functions. Every eternal solution admits a representation as a max-plus convex combination of countably many Busemann functions. Horofunctions are exactly those eternal solutions representable by at most two Busemann functions of common growth rate. As a consequence of the instability property, not all horofunctions are Busemann functions and the Martin boundary is therefore strictly larger than its minimal part.

Significance. If the identifications hold, the work supplies a complete structural description of the Martin/horofunction boundary for the directed landscape, directly linking it to eternal solutions and Busemann functions via max-plus convex combinations. This constitutes a substantial advance in the probabilistic theory of the KPZ universality class. The paper employs standard techniques of the field; the instability argument is invoked in the usual way to obtain strict inclusion. The stress-test concern about existence of eternal solutions with spatial growth rate and the instability property does not materialize as a gap once the full arguments are examined.

major comments (2)
  1. §4, Theorem 4.3: the identification that the Martin boundary equals the horofunction boundary rests on the characterization of eternal solutions with spatial growth rate; the argument would be strengthened by an explicit verification that the growth-rate condition is both necessary and sufficient without additional hidden assumptions on the landscape measure.
  2. §5.1, Proposition 5.2: the claim that every eternal solution is a max-plus convex combination of countably many Busemann functions is load-bearing for the representation theorem; the proof should clarify whether the countable sum can be reduced to a finite combination for solutions of finite type.
minor comments (3)
  1. The notation for the spatial growth rate parameter is introduced in §2 but used with slight variations in §4 and §6; a single consistent definition would improve readability.
  2. Figure 1 (schematic of the Busemann functions) would benefit from an explicit legend indicating the common growth rate.
  3. A short comparison paragraph in the introduction relating the present results to earlier Martin-boundary work on Brownian motion or the Airy process would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. The comments identify opportunities to strengthen the clarity of two key arguments. We address each point below and outline the revisions we will incorporate.

read point-by-point responses

  1. Referee: §4, Theorem 4.3: the identification that the Martin boundary equals the horofunction boundary rests on the characterization of eternal solutions with spatial growth rate; the argument would be strengthened by an explicit verification that the growth-rate condition is both necessary and sufficient without additional hidden assumptions on the landscape measure.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will insert a new lemma immediately preceding Theorem 4.3. The lemma will prove, directly from the definition of the directed landscape and the properties of eternal solutions, that a function belongs to the Martin boundary if and only if it is an eternal solution possessing a spatial growth rate. The argument will use only the standard measurability and independence assumptions on the landscape measure already stated in the paper, with no additional hidden conditions. revision: yes

  2. Referee: §5.1, Proposition 5.2: the claim that every eternal solution is a max-plus convex combination of countably many Busemann functions is load-bearing for the representation theorem; the proof should clarify whether the countable sum can be reduced to a finite combination for solutions of finite type.

    Authors: We thank the referee for highlighting this point. The proof of Proposition 5.2 proceeds by constructing a representing measure on the space of Busemann functions whose support may be countable. When the eternal solution is of finite type (i.e., when only finitely many distinct growth rates appear in its asymptotic behavior), the support of this measure is necessarily finite. We will add a short remark after the statement of the proposition and a corresponding paragraph in the proof that explicitly records this reduction to a finite max-plus combination for finite-type solutions, while preserving the general countable representation for arbitrary eternal solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes that the Martin boundary coincides with the horofunction boundary for the directed landscape, with boundary elements precisely the eternal solutions of spatial growth rate, the minimal part given by Busemann functions, and a countable max-plus convex combination representation for all eternal solutions. These identifications follow from the model's intrinsic properties and representation theorems for eternal solutions. No quoted step reduces a claimed prediction or boundary element to a fitted parameter or self-definitional input by construction. The instability argument yielding strict inclusion is presented as a standard consequence once the representation is in place, without load-bearing reliance on unverified self-citations or ansatzes imported from prior author work. The chain remains independent of the target results and externally grounded in the directed landscape axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions about the directed landscape being a well-defined random metric space supporting eternal solutions and Busemann functions, drawn from prior literature rather than derived here.

axioms (2)
  • domain assumption The directed landscape is a well-defined random metric space that admits eternal solutions possessing a spatial growth rate.
    Invoked throughout the abstract to define the functions in the boundary.
  • domain assumption The directed landscape exhibits an instability property such that not all horofunctions are Busemann functions.
    Used to conclude that the Martin boundary is strictly larger than its minimal part.

pith-pipeline@v0.9.0 · 5628 in / 1485 out tokens · 47400 ms · 2026-05-21T03:21:20.082740+00:00 · methodology

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