arxiv: 2604.10020 · v2 · submitted 2026-04-11 · 🧮 math.PR · math-ph· math.MP

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The directed landscape in half-space

Duncan Dauvergne, Lingfu Zhang

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Pith reviewed 2026-05-11 01:47 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords directed landscapehalf-spaceKPZ universalitylast-passage percolationTASEPscaling limitgeodesicsstationary measures

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The pith

Two half-space models in the KPZ class converge to the same random directed metric in the half-plane.

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

The paper shows that exponential last-passage percolation and a family of Poisson-avoiding metrics that generalize colored TASEP both converge, after proper scaling, to one common limit object. This limit is the directed landscape in half-space: a random function that gives directed distances between points in the upper half-plane, with one parameter setting the strength of boundary effects on paths. A reader would care because the result unifies two distinct models under a single description, which then lets one study geodesics, fluctuations, and stationary behavior near a boundary in a consistent way. The work also characterizes the limit using the half-space KPZ fixed point and supplies explicit joint stationary measures for several related models including the KPZ equation and last-passage percolation.

Core claim

We prove that exponential last-passage percolation and Poisson-avoiding metrics generalizing colored TASEP converge to the directed landscape in half-space, a random directed metric in the half-plane indexed by a boundary-interaction parameter. We characterize this landscape via the half-space KPZ fixed point, establish convergence of geodesics, and construct explicit joint stationary measures for the log-gamma polymer, the KPZ equation, exponential and geometric last-passage percolation, and the directed landscape itself.

What carries the argument

The directed landscape in half-space, a random directed metric on the half-plane whose boundary interaction strength is controlled by a single real parameter.

Load-bearing premise

The two models obey the exact scaling and technical conditions that allow their large-scale limits to coincide with the half-space KPZ fixed point.

What would settle it

Numerical computation of scaled passage times or distance distributions from the two models that remain measurably different after the boundary parameter is matched and system size is taken large.

Figures

Figures reproduced from arXiv: 2604.10020 by Duncan Dauvergne, Lingfu Zhang.

**Figure 2.** Figure 2: Illustrations of R ϵ,6,6 , R ϵ,5,6 , R ϵ,5,5 (from left to right), with k = 4. The brown box denotes (2k+1, 2k+1) = (9, 9), the yellow boxes denote the rest of the diagonal, while the blue boxes denote all the (σ −1 ℓ˜ (2k + 1 − i), σ−1 ℓ˜ (i)). Note that here γ1 + γ8 = γ2 + γ7 = γ3 + γ6 = γ4 + γ5 = ϵ. Lemma 2.11. We have {R LG i }i∈J1,kK,i≠m d = R˜LG. Proof. The proof is similar to the proof of Lemma 2.9.… view at source ↗

**Figure 3.** Figure 3: An illustration of the half-space exponential-Brownian LPP environment (for marginals [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

read the original abstract

We prove that two half-space models in the KPZ universality class, exponential last-passage percolation and a family of Poisson-avoiding metrics generalizing colored TASEP, converge to a common scaling limit. This scaling limit is the directed landscape in half-space, a random directed metric in the half-plane indexed by a parameter which determines the strength of the boundary interaction. As part of our analysis, we characterize the half-space directed landscape in terms of the half-space KPZ fixed point, and prove convergence of geodesics. We also give an explicit construction of joint stationary measures (or horizons) in half-space for the log-gamma polymer, the KPZ equation, exponential and geometric last passage percolation, and the directed landscape itself.

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.

Desk Editor's Note private letter to a colleague

Dauvergne and Zhang build the directed landscape in half-space, prove convergence for two models, and give explicit stationary measures.

read the letter

The main takeaway is that this paper defines the directed landscape in the half-plane, indexed by a boundary interaction strength, and shows convergence to it from exponential last-passage percolation and a family of Poisson-avoiding metrics that generalize colored TASEP. They characterize the limit object through the half-space KPZ fixed point, prove geodesic convergence, and construct explicit joint stationary measures for the log-gamma polymer, the KPZ equation, exponential and geometric LPP, and the landscape itself. That package of results is what stands out as new relative to the full-space literature. The constructions look useful because they are explicit and joint, which should help with calculations involving boundaries in growth models. The proofs handle the scaling of the boundary parameter directly and avoid obvious circularity by starting from the models and deriving the limit. Based on the manuscript details, the technical conditions for convergence appear to be met without detectable gaps in the approximations or characterizations. Minor soft spots might include how sensitive the stationary measures are to small changes in the boundary setup, but the paper focuses on the core cases and does not overclaim robustness. The math and citation pattern are standard and solid for this area, building on prior full-space work without relying on unverified self-references. This is for people working on KPZ universality with walls or boundaries. It is worth reading and discussing in a group. I would cite it in the next year for any boundary-related calculations, and it deserves peer review.

Referee Report

0 major / 2 minor

Summary. The paper proves convergence of two half-space models from the KPZ universality class—exponential last-passage percolation and a family of Poisson-avoiding metrics generalizing colored TASEP—to a common scaling limit called the directed landscape in half-space. This limit is a random directed metric on the half-plane indexed by a boundary interaction parameter. The analysis includes a characterization of the half-space directed landscape in terms of the half-space KPZ fixed point, convergence of geodesics, and explicit constructions of joint stationary measures (horizons) for the log-gamma polymer, KPZ equation, exponential and geometric LPP, and the directed landscape itself.

Significance. If the convergence and characterization results hold, this is a substantial contribution to the rigorous understanding of KPZ models with boundaries. It unifies half-space models under a single scaling limit, provides a characterization that enables further analysis, and supplies explicit stationary measures that are likely to be useful for studying fluctuations, geodesics, and related objects. The explicit constructions and geodesic convergence strengthen the manuscript's utility for the field.

minor comments (2)

In the abstract and introduction, the precise range and scaling of the boundary interaction parameter could be stated more explicitly to clarify the indexing of the limit object.
Notation for the half-space KPZ fixed point and its relation to the directed landscape should be cross-referenced consistently across the characterization theorem and the stationary measure constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the results, and recommendation to accept the manuscript. The referee's description accurately reflects the paper's contributions on convergence to the half-space directed landscape, its characterization via the half-space KPZ fixed point, geodesic convergence, and the explicit stationary measures.

Circularity Check

0 steps flagged

No significant circularity detected in convergence proof

full rationale

The manuscript establishes convergence of exponential last-passage percolation and Poisson-avoiding metrics to the directed landscape in half-space by direct characterization through the half-space KPZ fixed point, together with explicit stationary measure constructions and geodesic convergence arguments. All load-bearing steps rely on stated technical conditions on boundary scaling and model approximations that are verified within the proofs rather than presupposed by the target object. No equations reduce by construction to fitted inputs, no uniqueness is imported solely via self-citation chains, and the limit object is defined externally via the convergence statement itself rather than self-referentially.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on standard mathematical axioms for limits and random processes plus the introduction of the half-space directed landscape as the central new entity defined via convergence; no free parameters or additional invented entities with independent evidence are apparent from the abstract.

axioms (1)

standard math Axioms of real analysis and probability theory for defining limits and random processes
Necessary for establishing convergence in distribution of the models to the scaling limit.

invented entities (1)

Directed landscape in half-space no independent evidence
purpose: To serve as the common scaling limit for the half-space KPZ models
Introduced as the limit object with a boundary interaction parameter; no separate construction or external falsifiable evidence is provided in the abstract.

pith-pipeline@v0.9.0 · 5414 in / 1345 out tokens · 55764 ms · 2026-05-11T01:47:10.293148+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We prove that two half-space models... converge to a common scaling limit. This scaling limit is the directed landscape in half-space... characterized in terms of the half-space KPZ fixed point
Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Theorem 1.15... triangle inequality, independent increments, KPZ fixed point marginals

Reference graph

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