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arxiv: 2605.22787 · v1 · pith:LVKXUNOYnew · submitted 2026-05-21 · 🧮 math.PR

Invariant measures for half-space geometric LPP: classification and the one force--one solution principle

Sayan Das , Evan Sorensen , Zongrui Yang This is my paper

Pith reviewed 2026-05-22 03:00 UTC · model grok-4.3

classification 🧮 math.PR
keywords invariant measureshalf-space last-passage percolationgeometric LPPKPZ universalityBusemann processone force one solutionsemi-infinite geodesicsboundary effects
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The pith

Half-space geometric last-passage percolation has all its extremal invariant measures classified by asymptotic slope, including a gap under strong boundary attraction.

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

The paper establishes a complete list of extremal invariant measures for this boundary model by proving that every initial condition with a prescribed far-away slope relaxes to a unique limiting process. A reader cares because the result is the first full classification for any KPZ-class model that combines boundary effects with an unbounded spatial domain. The authors introduce a one-force-one-solution principle: the recentered height function at the present time, started from the distant past, converges in distribution to the Busemann process associated with the chosen slope. This holds uniformly across all slopes, even when the boundary is strongly attractive and forces a jump in the set of allowed slopes. The same argument also shows that the previously constructed family of jointly invariant measures is in fact extremal.

Core claim

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is achieved by establishing a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at infinity, the recentered solution at time zero converges to a process distributed as the associated invariant measure with the specified slope, called the Busemann process. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP.

What carries the argument

The one force--one solution principle, which shows that any initial condition with fixed asymptotic slope at infinity converges to a unique limiting process (the Busemann process) determined solely by that slope.

If this is right

  • The extremal invariant measures are exactly the Busemann processes indexed by admissible slopes.
  • Under strong attractive boundary the admissible slopes form a set with a discontinuity that is absent in the full-space model.
  • The directions of all semi-infinite geodesics are classified, confirming an earlier conjecture.
  • The jointly invariant measures constructed in prior work are extremal.

Where Pith is reading between the lines

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

  • Similar convergence arguments may classify invariant measures for other half-space growth models such as PNG or TASEP with boundary.
  • Numerical checks of geodesic direction statistics could test the predicted slope discontinuity directly.
  • The boundary-induced gap in slopes may produce distinct fluctuation exponents near the wall that are worth isolating in simulations.

Load-bearing premise

Directions of semi-infinite geodesics can be controlled sufficiently well to handle the jump in admissible slopes created by a strongly attractive boundary.

What would settle it

A concrete initial condition with a fixed far-away slope whose recentered evolution at time zero fails to match the law of the corresponding Busemann process.

Figures

Figures reproduced from arXiv: 2605.22787 by Evan Sorensen, Sayan Das, Zongrui Yang.

Figure 1
Figure 1. Figure 1: An up-right path in half-space. Paths are confined to the right of the diagonal line. In the case of geometric half-space LPP, The weights on the black vertices along the diagonal have the Geo(cq) distribution, while the weights on the gray vertices in the bulk have the Geo(q 2 ) distribution. We call this model half-space last-passage percolation (LPP). A geodesic is a (possibly non-unique) maxi￾mizing pa… view at source ↗
Figure 2
Figure 2. Figure 2: Phase diagram for the collection of invariant measures in half-space geometric LPP Define an environment of independent random variables (Y ε (i,j) )(i,j)∈Z 2 HS as follows: for i ∈ N, let Y ε (i,i) ∼ Geo(qsi), and for integers i > j, set Y ε (i,j) ∼ Geo(sisj ). Let G ε denote half-space last-passage percolation where the variables ω in (1.1) are replaced by Y ε . We use the notation G ε to distinguish fro… view at source ↗
Figure 3
Figure 3. Figure 3: A path from (m, ℓ) to (n, n) (blue, thick) and its image (orange, thin) across the dotted line y = n − x. 2.4. Eternal solutions. We say that a function b : Z 2 HS → R is an eternal solution if, for all m, n ∈ Z, b(m, n) = ( ω(m,n) + b(m, n − 1) ∨ b(m − 1, n), m > n ω(n,n) + b(n, n − 1), m = n. (2.4) For any initial condition f : Z≥j , the recursion for G (1.2) the function Gf,j satisfies this recursion fo… view at source ↗
Figure 4
Figure 4. Figure 4: The geodesics γ (blue, thick) and π (orange, thin) both terminate at (m0, n0), and when moving downwards, there is a first place x−p = (m−p, n−p) where the two geodesics split. Proof. Items (i)-(ii): Observe that (m0, n0) = (m, n). By Lemma 2.8 and definition of the path γ(m,n) (2.8), we have ω(m−p,n−p) = b(m−p, n−p) − b(m−(p+1), n−(p+1)), for all p ∈ Z≥0. (2.9) Then, for every k ∈ Z≥0, X k p=0 ω(m−p,n−p) … view at source ↗
Figure 5
Figure 5. Figure 5: The geodesic (blue/thick) from (−n + ξn, −n) to (k, k) will first travel to the boundary (black, thin) at approximately the point (−xn, −xn), then move along the diag￾onal to (k, k). The blue path is shown to be slightly offset from the boundary for visual purposes. The gray shading indicates the half-space Z 2 HS. and so by the definition of ρc in (3.1), the function κ 7→ ρc(κ) is linear on the interval h… view at source ↗
Figure 6
Figure 6. Figure 6: A depiction of the argument that, if An holds, then Ap fails for all p sufficiently less than n. Dashed lines with slope 1 are drawn from the points (n + m, n) and (n + k, n). Since the geodesic γ(n+k,n) has inverse slope less than 1, the geodesic γ(n+m,n) eventually lies to the right of the point (p + k, p). By ordering of geodesics, the inverse-slope of γ(p+k,p) is bounded below by the direction of γ(n+m… view at source ↗
read the original abstract

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.

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

1 major / 1 minor

Summary. The manuscript proves a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with arbitrary boundary parameter. It establishes a one force--one solution principle showing that solutions started from arbitrary initial conditions with given asymptotic slope at infinity converge to the associated Busemann process (the first such construction for a half-space model). This confirms the classification conjectures of Barraquand-Corwin and the joint invariance conjecture of Dauvergne-Zhang. The argument classifies directions of semi-infinite geodesics and combines this control with half-space Gibbsian line ensemble techniques to handle the discontinuity in admissible slopes that arises under a strong attractive boundary.

Significance. If the central claims hold, the work is a substantial advance: it supplies the first complete extremal classification for any KPZ-class model that incorporates both boundary effects and an unbounded domain. The construction of the half-space Busemann process and the independent verification of the joint invariance conjecture add concrete new objects and confirmations to the literature on invariant measures and geodesic behavior in the KPZ universality class.

major comments (1)
  1. [proof of the one force--one solution principle and geodesic direction classification] The classification of semi-infinite geodesic directions (used to resolve the slope discontinuity for strong attractive boundaries) is load-bearing for the one force--one solution principle. The manuscript combines this classification with half-space Gibbsian line ensemble techniques, but the direct applicability of the line-ensemble hypotheses to the geometric weight distribution is asserted rather than verified in detail; an explicit check that the required regularity and Gibbsian properties hold for the geometric case would be needed to guarantee that the limiting object coincides with the conjectured invariant measure.
minor comments (1)
  1. [Introduction and Section 2] Notation for the boundary parameter and the admissible slope sets could be made more uniform across the introduction and the main statements to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their positive assessment of the results on the classification of extremal invariant measures and the one force--one solution principle. We address the major comment in detail below.

read point-by-point responses

  1. Referee: [proof of the one force--one solution principle and geodesic direction classification] The classification of semi-infinite geodesic directions (used to resolve the slope discontinuity for strong attractive boundaries) is load-bearing for the one force--one solution principle. The manuscript combines this classification with half-space Gibbsian line ensemble techniques, but the direct applicability of the line-ensemble hypotheses to the geometric weight distribution is asserted rather than verified in detail; an explicit check that the required regularity and Gibbsian properties hold for the geometric case would be needed to guarantee that the limiting object coincides with the conjectured invariant measure.

    Authors: We agree that the classification of semi-infinite geodesic directions is central to resolving the slope discontinuity under strong attractive boundaries and forms a key step in the proof of the one force--one solution principle. This classification is established in Theorem 3.1 and the analysis of Section 3, confirming the conjecture of Dauvergne and Zhang. Regarding the half-space Gibbsian line ensemble techniques, the manuscript invokes the general framework from the literature on Gibbsian line ensembles, which applies to weight distributions satisfying standard regularity conditions (finite moments, suitable tail decay, and the Gibbs property). While these conditions are satisfied by the geometric distribution and are implicitly verified through the explicit computations in the geometric LPP setting, we acknowledge that a more detailed, self-contained check would improve clarity. In the revised manuscript we will add an explicit verification subsection (new Section 4.3) that directly confirms the required regularity and Gibbsian properties for the geometric weights, thereby guaranteeing that the limiting object coincides with the conjectured invariant measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation supplies independent limiting construction

full rationale

The paper derives the complete classification of extremal invariant measures by proving a one force--one solution principle that constructs the Busemann process as the limit of recentered solutions started from arbitrary initial conditions with prescribed asymptotic slope. This limiting argument is presented as supplying the required control on semi-infinite geodesic directions and applicability of half-space Gibbsian line ensemble methods to geometric LPP, thereby confirming (rather than presupposing) the conjectures of Barraquand-Corwin and Dauvergne-Zhang. No step reduces by definition, by renaming a fitted quantity as a prediction, or by a load-bearing self-citation chain; the central claims rest on the new limiting result and the cited external techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

With only the abstract available, a full ledger cannot be extracted; the work relies on standard background results in probability theory and on prior constructions of invariant measures and line ensembles whose validity is assumed rather than re-derived here.

axioms (2)
  • domain assumption Existence and basic properties of semi-infinite geodesics in half-space geometric LPP
    Invoked to control directions and handle the slope discontinuity under strong boundary attraction
  • domain assumption Applicability of half-space Gibbsian line ensemble techniques to the geometric LPP setting
    Used to obtain the necessary tightness and convergence statements

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