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Permutation invariance in last-passage percolation and the distribution of the Busemann process

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Pith's one-line read For exponential last-passage percolation, the joint distribution of all Busemann increments in any finite grid and any finite set of directions equals the joint distribution of last-passage increments in a slightly larger finite grid with…

desk verdict Genuinely new finite-dimensional description of the Busemann process in exponential LPP, proved in detail; the main risk is its heavy but transparent reliance on an overlapping preprint's inhomogeneous Busemann theory. read the letter →

arxiv 2506.12641 v2 pith:VJNQ3S5H submitted 2025-06-14 math.PR

Erik Bates , Elnur Emrah , James Martin , Timo Seppäläinen , Evan Sorensen This is my paper · ORCID
classification math.PR MSC 60K3560K3760K25
keywords last-passagepercolationBusemannfunctionsexponentialweightspermutationinvarianceBurkepropertyqueuinginterpretationinhomogeneousenvironmentjointdistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the complicated joint law of Busemann functions in exponential last-passage percolation can be described exactly by a finite inhomogeneous environment. Busemann functions are limiting differences of growth times as the terminal point moves to infinity in a fixed direction. The authors show that, inside any k by l grid and for any d directions, all Busemann increments are equal in joint distribution to last-passage increments in a (k+d-1) by (l+d-1) grid, terminating at d points on an antidiagonal. Because the description uses only finitely many independent exponential variables, it gives a concrete way to sample the joint Busemann distribution. The proof introduces a new permutation invariance of inhomogeneous last-passage times, proved through the Burke property rather than through RSK or Schur-function formulas.

What carries the argument

The central device is a new permutation invariance (Theorem 3.1): the joint law of inhomogeneous last-passage times between certain endpoint pairs is preserved when the inhomogeneity parameters of columns and rows are permuted. It is proved by swapping two neighboring rows or columns through an explicit coupling based on a queue with exponential service and arrivals, where the Burke property shows that the unused-service transformation swaps the rate parameters while preserving all crossing passage times. The second key ingredient is the theory of thin Busemann functions, column- or row-limited versions of Busemann limits in inhomogeneous exponential LPP, together with an induced-weights identity that rewrites increments to far-off terminal points as increments in a smaller finite grid. These ingredients combine to convert the infinite Busemann joint law into a finite-grid last-passage problem whose weights are explicitly described by (2.13).

What would settle it

Choose a small explicit case such as k=2, l=2, d=2 with directions r_1 < r_2. Theorem 2.4 expresses the joint law of the four Busemann increments as explicit functions of five independent exponentials; compute that joint law numerically, then simulate long last-passage paths to terminals (m_1,n) and (m_2,n) with m_1/n to r_1 and m_2/n to r_2, and compare the empirical joint distribution with the predicted finite formula. Any systematic mismatch in the joint CDF would refute the theorem.

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Extended reading notes

Core claim

Theorem 2.4 states that the collection of Busemann increments ($I^{{r_p}}$_u, $J^{{r_p}}$_v) for u in the horizontal-edge set, v in the vertical-edge set, and directions r_1 < ... < r_d is distributionally identical to the collection of last-passage increments (I_{u,z_p}[\eta], J_{v,z_p}[\eta]) computed in a finite inhomogeneous environment \eta with independent exponential weights of rates a_i + b_j, where the sequences a and b are built from the values \zeta(r_p) and the terminal points z_p lie on an antidiagonal of the enlarged grid. ВThus a genuinely infinite-dimensional object, the Busemann process across both space and direction, is encoded by finitely many random variables. The paper also derives from this identity a complete characterization of the Busemann process on a single lattice edge, recovers an independence theorem of Shen in a special case, and extends the description to include the axis directions.

Load-bearing premise

The derivation takes as given a body of results on Busemann functions in inhomogeneous exponential last-passage percolation from a related preprint by overlapping authors, especially the identification of directional Busemann limits with thin column and row limits; if those results had narrower validity than assumed, the proof of the main theorem would not go through.

Editorial extensions

If this is right

  • Every finite joint distribution of Busemann increments for any set of edges and directions can be sampled exactly from finitely many independent exponentials, without simulating an infinite environment.
  • The Busemann process on a single lattice edge has independent increments with respect to direction, and the distribution of each increment is explicitly computable, recovering and reproving known results without queuing maps.
  • The description extends to the axis directions r = 0 and r = 8 by a simple two-sided version that also records the underlying i.i.d. weights.
  • A special case of Shen's independence theorem, concerning monotone variation of direction along a down-right path, follows directly from the finite-grid representation.
  • The permutation invariance provides an explicit coupling of the weights before and after swapping inhomogeneity parameters, giving a tool that may apply to other problems involving inhomogeneous last-passage percolation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: Because the finite environment uses only finitely many random variables, the result gives a practical numerical route to previously inaccessible quantities such as multi-direction geodesic coalescence probabilities inside a finite box.
  • Editorial: The coupling-based proof of permutation invariance suggests that analogous invariance may hold for directed polymers at positive temperature, where arrival and service are replaced by ratios of partition functions and a Burke-type stationarity is available.
  • Editorial: Combining this finite representation with scaling limits such as the directed landscape could lead to testable approximations: the joint law of Busemann increments in a growing box should converge to the corresponding multi-direction quantities in the continuum scaling limit.
  • Editorial: The multi-point version mentioned in the paper indicates that the invariance is not special to single paths, so extensions to multi-point last-passage observables may allow exact finite sampling of more complex functionals.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper studies the joint distribution of Busemann functions in i.i.d. exponential last-passage percolation. The main result, Theorem 2.4, asserts that Busemann increments on a k x l grid for d directions are equal in distribution to point-to-point last-passage increments in a finite inhomogeneous environment on a (k+d-1) x (l+d-1) grid with terminal points along an antidiagonal. The proof introduces a new permutation invariance of inhomogeneous LPP (Theorem 3.1) proved via a queueing/Burke argument, and imports the inhomogeneous Busemann theory from the preprint [24]. The paper also derives corollaries on independent increments and Shen's independence theorem.

Significance. If the central identity is correct, it gives the first explicit finite-dimensional description of the joint Busemann distribution for arbitrary edge sets, which was previously available only along a horizontal line. The permutation invariance theorem is of independent interest and its proof via the Burke property is elegant and distinct from RSK-based approaches. The paper is clearly written, with a detailed two-direction proof sketch and explicit examples that make the result accessible. The main caveats are the incomplete proof of the full permutation invariance stated in Theorem 3.1 and the heavy reliance on an unreviewed preprint by overlapping authors.

major comments (2)
  1. [Section 4, Theorem 3.1] The proof of Theorem 3.1 is incomplete as stated. The two-row swap Propositions 4.1 and 4.2 are proved only under the assumption b2 > b1 (see the setup before (4.2) and Lemma 4.5, where the stationary distribution Exptb2-b1u requires b2>b1). The theorem, however, allows arbitrary finite permutations with no ordering condition, and the proof of Theorem 3.1 does not explain how to handle the opposite order for rows or the analogous ordering condition for columns. This is not a cosmetic gap: in Lemma 6.1, for example, the invariance is used to interchange a column of rate 1-zeta(r_p) with a column of rate 1, with the smaller-rate column to the right of the larger-rate column, which is exactly the case excluded by the transposed-row condition a_{i+1} > a_i. The authors should either prove the missing cases (or show they follow from the proved case by a limiting argument) or restrict Theorem 3.1 to the situations actually verified and used.
  2. [Sections 5-6] The proof of Theorem 2.4 relies crucially on Propositions 5.1, 5.3, 5.4, 5.5 and Corollary 5.2, which are imported from the preprint [24] and not proved in this manuscript. In particular, the identification at the critical direction in (6.28) uses Proposition 5.5 at the endpoint r = c_i^ver; the proof would fail if that proposition requires strict inequality. Since [24] is a preprint by overlapping authors and the presented results are load-bearing for the central identity, the authors should include a self-contained proof of the necessary statements (at least the endpoint case of Proposition 5.5) or, if that is impractical, clearly state the dependency and verify explicitly that all hypotheses of the imported results are satisfied in every application in Section 6.
minor comments (2)
  1. [Section 6, Lemma 6.1] In the proof of Lemma 6.1, equation (6.14) is presented as a chain of equalities and an inequality, but the first 'equality' appears to combine a re-indexing of the event with a monotonicity step; the logical structure should be clarified for readability.
  2. [Section 2, equations (2.3)-(2.4)] The same symbols I and J are used for initial-point and terminal-point increments; while this follows the field's conventions, the double use alongside Busemann functions I^r and J^r may be confusing on first reading. A brief remark or a change of notation for terminal increments could help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the joint-distribution identity is proved by comparing joint CDFs, not built into the definition of the eta environment.

full rationale

Theorem 2.4 is not circular. The eta rates in (2.13) are chosen from the one-directional rate function zeta(r) of Proposition 2.2(b), but the joint identity is not imposed: Section 6 proves it by bounding the prelimit probabilities in (6.4), using the permutation invariance of Theorem 3.1 (proved self-contained in Section 4 via the Burke property) and the thin-Busemann identifications of Section 5. The imported results from [24] (Propositions 5.1, 5.3, 5.4, 5.5 and Corollary 5.2) are external statements with the stated hypotheses (5.1)-(5.2), and none of them assumes Theorem 2.4. The critical endpoint use at (6.28) is exactly the case r = c_i^ver allowed by Proposition 5.5; this is a dependency on an overlapping-authors preprint, hence a provenance and verification risk, but it is not a definitional reduction. Proposition 2.2 is also used only to provide the Busemann limits and continuity in (6.3)-(6.4); Remark 2.7's later derivation of the d=1 marginals from Theorem 2.4 is a consistency check, not an input to the proof. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force the choice, and no known result is merely relabeled. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on the standard LPP framework, the inhomogeneous Busemann theory of [24], the law of large numbers of [22], and the new permutation invariance proved via the Burke property. No parameters are fitted to data; the eta rates are deterministic functions of the input directions. No new physical entities are posited.

assumptions (5)
  • domain assumption Existence, marginal distributions, and down-right path independence of Busemann functions in i.i.d. exponential LPP (Proposition 2.2), cited from [56] Theorem 4.2 and [11] Lemma 3.3.
    Used to start the proof and to justify replacing Busemann functions by limits of finite-grid increments in (6.3)-(6.4).
  • domain assumption Inhomogeneous Busemann functions satisfy Propositions 5.1, 5.3, 5.4, 5.5 and Corollary 5.2 of [24] under conditions (5.1)-(5.2).
    These results provide thin Busemann marginal distributions, independence along down-right paths, and the identification of directional limits with thin Busemann limits; central to Lemmas 6.1, 6.2 and 6.5.
  • domain assumption Law of large numbers for inhomogeneous LPP (Proposition 6.3, from [22, Theorem 3.7]).
    Used in Lemma 6.4 to compare growth rates and ensure geodesics enter from the top row for large n.
  • standard math The queueing Burke property as derived in Lemma 4.5 (proved in this paper).
    This is the engine behind the permutation invariance Theorem 3.1, which is invoked repeatedly in Section 6.
  • standard math Lemma 2.1 monotonicity of increments, from [54, Lemma 6.2] and [56, Lemma 4.6].
    Used for increment comparisons when shifting endpoints in Lemmas 6.1 and the proof sketch.

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Pith. "Pith review of Permutation invariance in last-passage percolation and the distribution of the Busemann process." pith.science (2026). https://pith.science/paper/VJNQ3S5H

@misc{pith2026250612641,
  author       = {Pith},
  title        = {Pith review of: Permutation invariance in last-passage percolation and the distribution of the Busemann process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJNQ3S5H}},
  note         = {Machine review of arXiv:2506.12641}
}
abstract

In i.i.d. exponential last-passage percolation, we describe the joint distribution of Busemann functions, over all edges and over all directions, in terms of a joint last-passage problem in a finite inhomogeneous environment. More specifically, the Busemann increments within a $k\times\ell$ grid, and associated to $d$ different directions, are equal in distribution to a particular collection of last-passage increments inside a $(k+d-1)\times(\ell+d-1)$ grid. The joint Busemann distribution was previously described along a horizontal line by Fan and the fourth author, using certain queuing maps. By contrast, our new description explicitly gives the joint distribution for any collection of edges (not just along a horizontal line) using only finitely many random variables. Our result thus provides an exact and accessible way to sample from the joint distribution. In the proof, we rely on one-directional marginal distributions of the inhomogeneous Busemann functions recently studied by Janjigian and the second and fourth authors. The second ingredient of our proof is a novel joint invariance of inhomogeneous last-passage times under permutations of the inhomogeneity parameters. Our proof of the invariance is different from earlier proofs of such results, using the Burke property instead of the RSK correspondence, and leading to an explicit coupling of the weights before and after the permutation of the parameters.

Figures

Figures reproduced from arXiv: 2506.12641 by the authors.

Figure 2.1
Figure 2.1. Illustration of the geometric arrangement in Theorem 2.4 with d “ 4, k “ 6, and ℓ “ 4. Theorem 2.4 gives the distribution of the Busemann functions in the rks ˆ rℓs grid with lower left corner at p1, 1q and upper right corner at pk, ℓq, by considering inhomogeneous exponential LPP terminating at the vertices z1, . . . , zd. Outside of the shaded region, the weight at vertex pi, jq is exponential with rate ai `bj , w… view at source ↗
Figure 2.2
Figure 2.2. Illustration of the conclusion of Theorem 2.4 with d “ 3. The following two collections of random variables are equal in distribution. (a) The Busemann increments within rks ˆ rℓs for directions r1 ă r2 ă r3 (red, blue, and teal arrows). These increments depend on infinitely many variables in the i.i.d. environment ω. (b) The last-passage increments (with respect to initial points within rks ˆ rℓs) to the terminal p… view at source ↗
Figure 2.3
Figure 2.3. Collapsing of [PITH_FULL_IMAGE:figures/full_fig_p010_2_3.png] view at source ↗
Figures from the paper (11 more)
Figure 2.4
Figure 2.4. Figure 2.4: The η-weights on the grid rp1, 1q, zps form a stationary inhomogeneous LPP model for each p P rds. In the picture, p “ 2. Remark 2.9 (Monotonicity of Busemann functions). As mentioned in Section 1.A, a key feature of Busemann functions is monotonicity with respect to…
Figure 2.5
Figure 2.5. Figure 2.5: Special case of [PITH_FULL_IMAGE:figures/full_fig_p011_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Special case of [PITH_FULL_IMAGE:figures/full_fig_p012_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Illustrates the argument in (2.40). The goal is to introduce inhomo￾geneities into the environment. The white vertices are located at pk 1 n , nq and pk 2 n , nq, and have limiting directions r1 ă r2. The black vertices are located at pk 1 n ` 1, nq and pk 2 n ` 2, n…
Figure 2.8
Figure 2.8. Figure 2.8: Illustrates the steps from (2.41) to (2.48). The goal is to replace the increments from [PITH_FULL_IMAGE:figures/full_fig_p019_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Illustrates (2.49)–(2.61). The goal is to identify thin Busemann func￾tions/their prelimits (solid arrows) with finite grid LPP increments (dashed arrows). Through (2.49), one can rewrite the probability on the right-hand side of (2.48) as PtI k,Ò u rω 1 s ą xu,1 and…
Figure 2.10
Figure 2.10. Figure 2.10: Illustrates the rest of the proof after (2.61). The goal is to send ϵ Ñ 0. A black dot indicates an Exptϵu-weight (tending to 8), while a white dot indicates a zero weight [PITH_FULL_IMAGE:figures/full_fig_p024_2_10.png]
Figure 3.1
Figure 3.1. Figure 3.1: Illustrates an application of Theorem 3.1 on the finite grid rms ˆ rns with m “ 13 and n “ 11. By the action of σ, the parameters of columns 9 and 10 (orange) are interchanged, and the parameters of columns 3, 4, and 5 (green) are rearranged in some way. By the actio…
Figure 4.1
Figure 4.1. Figure 4.1: The green (upper) path has weight Kx i,j and the blue (lower) path has weight Kx 1 i,j . The difference in their weights is the sum of the difference terms cs (indicated in red). We will fix i and establish the conclusion of Proposition 4.1 by induction on j. Let x0 …
Figure 4.2
Figure 4.2. Figure 4.2: Illustration of two cases of a single time-step of the queue. The initial queue-length is Q. Service S is available, followed by an arrival A. The queue-length at the end of the step is Q˜. On the left, the case S ď Q, where U “ 0 and D “ S. On the right, the case S …
Figure 5.1
Figure 5.1. Figure 5.1: Illustrates the thin Busemann functions I k,Ò p1,1q rws and I ℓ,Ñ p1,1q rws. The last-passage times Lp1,1q,pk,nq rws and Lp2,1q,pk,nq rws are indicated with red arrows. The last-passage times Lp1,1q,pm,ℓq rws and Lp2,1q,pm,ℓq rws are indicated with blue arrows. J ℓ,Ñ…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.