{"id":"afb28071-a835-473d-afe1-63547e3a8cfd","arxiv_id":"2506.12641","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The joint law of Busemann increments in i.i.d. exponential LPP is exactly represented by last-passage increments on a finite grid with inhomogeneous exponential weights.","lead":"In exponential last-passage percolation, the joint distribution of Busemann functions across all edges and all directions is shown to equal the distribution of last-passage increments in a small inhomogeneous random grid. This gives an exact finite way to sample a central object of the KPZ universality class, previously described only along a single horizontal line.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central identity is conditional on the inhomogeneous Busemann theory of [24]; Proposition 5.5, used at the critical direction in (6.28), is not reproved, and if it has narrower validity the Section 6 derivation collapses.","rationale":"The reader's weakest assumption identifies exactly the load-bearing concern: the proof of Theorem 2.4 leans on the inhomogeneous Busemann theory of [24], especially Proposition 5.5 at the endpoint of the critical interval. I reviewed the derivation in Section 6 and confirmed that (6.28) is a pivotal step in Lemma 6.2, and that Lemma 6.5 continues to use [24, Prop. 5.1 and 5.3] to identify the limit weights. Since [24] is a preprint by overlapping authors and is not reproved, the central claim is conditional on its correctness. This is not an ad hominem or a consensus objection; it is a precise statement about where the proof is least self-contained. I do not find a fatal internal inconsistency: the row-swap ordering gap in Theorem 3.1 is fixable by applying the proved direction to the swapped parameters, and the atom-related limit issues are resolved by the continuity of LPP increments and by monotone approximation. The paper is a major advance, and the CONDITIONAL verdict with MODERATE confidence is appropriate. My stress-test does not change that verdict, so verdict_should_be is UNCHANGED.","tokens_in":54141,"tokens_out":18232,"duration_ms":222157,"concrete_test":"Verify the specific instance of Proposition 5.5 used at (6.28): for parameters a_i = 0 for i < k0(q), a_{k0(p)} = -zeta(r_p) for p <= q, and a_i = 0 otherwise, with b_j = 1, independently derive that the directional Busemann limit I^{r_q}_{(i,j)} equals the thin Busemann limit I^{k0(q),↑}_{(i,j)} for all (i,j) in [k] x [l], using only the homogeneous Busemann theory (Proposition 2.2) and the explicit finite-inhomogeneity structure. If this restricted equality can be proved, the most delicate external invocation is validated; if not, Lemma 6.2 has a genuine hole and Theorem 2.4 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.4 is proved by bounding the prelimit probability in (6.4) and passing to limits. Lemma 6.2 performs this passage using equation (6.28), which identifies the directional Busemann limit I^{r_q} with the thin Busemann limit I^{k0(q),↑} exactly at the critical direction r_q = c_i^ver, computed in (6.26) from the formulas in [24]. This identification is an endpoint case of Proposition 5.5, one of several results (Propositions 5.1, 5.3, 5.4, 5.5 and Corollary 5.2) imported without proof from the preprint [24] by overlapping authors. The same imported theory drives the later, equally essential identification in Lemma 6.5, where the marginals of the limiting induced weights are read off from Proposition 5.1 and independence from Proposition 5.3. If any of these imported statements is incorrect or has hypotheses narrower than the cases invoked here—for example, if Proposition 5.5 requires strict inequality r < c_i^ver instead of r ≤ c_i^ver—then the inequalities in (6.32) and the final equality of Lemma 6.5 do not follow, and Theorem 2.4 is unproven in the form stated. The paper offers no independent derivation, no machine-checked proof, and no numerical corroboration of this external dependency. This is the single most load-bearing assumption because it is the only non-self-contained component of the proof; all other ingredients, including Theorem 3.1, Lemma 6.1, Lemma 6.4, and Proposition 6.3, are either proved in the paper or are standard. The row-swap ordering issue in the proof of Theorem 3.1 is real but readily fixable by symmetry, so it is less load-bearing than the imported Busemann theory.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":54478,"tokens_out":11349,"duration_ms":140742,"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":[{"comment":"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.","section":"Section 4, Theorem 3.1"},{"comment":"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.","section":"Sections 5-6"}],"minor_comments":[{"comment":"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.","section":"Section 6, Lemma 6.1"},{"comment":"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.","section":"Section 2, equations (2.3)-(2.4)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically ambitious and the main theorem is likely correct, but the proof has two structural weaknesses: an incomplete proof of Theorem 3.1 as stated, and a heavy dependence on an unreviewed preprint by overlapping authors. The editor may wish to consider whether the paper should be accepted only after the status of [24] is clarified, or whether the authors should be asked to provide proofs of the imported statements in an appendix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what it promises: Theorem 2.4 gives the joint distribution of Busemann increments across all edges of a finite grid and all directions, as last-passage increments in a finite inhomogeneous environment. That is a real advance over Fan–Seppäläinen's horizontal-line description, and it comes with clean corollaries (independent increments, a short proof of Shen's theorem). The proof strategy is also new: the permutation invariance in Theorem 3.1 is proved via an explicit coupling and the Burke property, rather than RSK or Schur machinery. I found no fatal flaw in the argument as sketched and as written in the lemmas.\n\nThe soft spots are proportionate. The proof leans heavily on the inhomogeneous Busemann theory of Emrah–Janjigian–Seppäläinen [24], a preprint by overlapping authors. That is a genuine dependency, and if any of its key propositions fails, the Section 6 derivation collapses. The stress-test note worried about Proposition 5.5 being used at the endpoint r = c_i^ver; on reading the paper, Proposition 5.5 is stated for the closed interval [0, c_i^ver], so the endpoint is covered. The concern only bites if [24] itself is wrong, not because of a misuse here. Still, the authors do not reprove these imports, and the paper would be more robust if they did.\n\nThe second issue is minor: the row-swap proof in Section 4 assumes b2 > b1 for the queueing stability, while Theorem 3.1 is stated for all permutations. The reverse ordering is handled by symmetry or by running the queue the other way, but it is not spelled out. Easy fix, not a structural problem.\n\nI did not line-by-line verify all 55 pages, but the structure is coherent, the lemmas are checkable, and the central identity is not assumed. This is a significant paper for the KPZ/LPP community.\n\nRecommendation: send it to serious peer review. Ask a referee specifically to check the imported results from [24] and the row-swap case in Theorem 3.1. With those verified, the paper should be accepted.","headline":"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.","tokens_in":55084,"tokens_out":1801,"would_cite":true,"duration_ms":23265,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60K37","60K25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["last-passage percolation","Busemann functions","exponential weights","permutation invariance","Burke property","queuing interpretation","inhomogeneous environment","joint distribution"],"falsifier":"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.","tokens_in":53921,"feed_emoji":"📐","tokens_out":4189,"duration_ms":55621,"temperature":0.7,"pith_summary":"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.","feed_headline":"Busemann joint law sampled from a finite grid","feed_subtitle":"For any k-by-l box and d directions, the joint law equals increments to d points in a larger inhomogeneous grid.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the inhomogeneous Busemann function theory, including thin Busemann function limits, marginals, and independence properties, that the proof imports wholesale.","marker":"[24]"},{"why":"Provides the increment monotonicity lemma used throughout and the induced-weights identity that carries the reduction to finite grids.","marker":"[56]"},{"why":"Records the classical distributional formula for inhomogeneous last-passage times, the context from which the new invariance generalizes.","marker":"[7]"},{"why":"States a related hidden invariance of last-passage percolation and directed polymers, used to calibrate the scope of the new permutation invariance.","marker":"[18]"},{"why":"Establishes interchangeability of exponential queues in series, the queueing principle behind the two-row coupling argument.","marker":"[62]"},{"why":"Presents the unused-service coupling idea that the proof of the Burke property for the two-row swap uses.","marker":"[61]"},{"why":"Provides the previously proved independence property of Busemann functions that the paper recovers as a corollary of its finite-grid description.","marker":"[60]"},{"why":"Gives the positive-temperature analogue of the coupled stationary multi-layer environment, identifying the finite inhomogeneous structure as the stationary multi-path inverse-gamma polymer.","marker":"[12]"}],"fun_headline_variants":["Finite grid encodes full Busemann joint law","Busemann law on any grid from a finite sample","Joint Busemann law via finite inhomogeneous grid","Permutation invariance yields finite Busemann sampling","Busemann process sampled from finite weighted grid"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Finite grid encodes full Busemann joint law","Busemann law on any grid from a finite sample","Joint Busemann law via finite inhomogeneous grid","Permutation invariance yields finite Busemann sampling","Busemann process sampled from finite weighted grid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1565,"prompt_tokens":1002,"completion_tokens":563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":493}},"tokens_in":618,"tokens_out":563,"duration_ms":6270,"temperature":1.0,"reasoning_tokens":493,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:46:07.292505+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights","cited_arxiv_id":"1709.05771","evidence_quote":"Provides the increment monotonicity lemma used throughout and the induced-weights identity that carries the reduction to finite grids."},{"cited_title":"Borodin and S","cited_arxiv_id":null,"evidence_quote":"Records the classical distributional formula for inhomogeneous last-passage times, the context from which the new invariance generalizes."},{"cited_title":"Dauvergne","cited_arxiv_id":null,"evidence_quote":"States a related hidden invariance of last-passage percolation and directed polymers, used to calibrate the scope of the new permutation invariance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes interchangeability of exponential queues in series, the queueing principle behind the two-row coupling argument."},{"cited_title":"Tsoucas and J","cited_arxiv_id":null,"evidence_quote":"Presents the unused-service coupling idea that the proof of the Burke property for the two-row swap uses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the previously proved independence property of Busemann functions that the paper recovers as a corollary of its finite-grid description."},{"cited_title":"Chaumont","cited_arxiv_id":null,"evidence_quote":"Gives the positive-temperature analogue of the coupled stationary multi-layer environment, identifying the finite inhomogeneous structure as the stationary multi-path inverse-gamma polymer."}],"review_version":1}