{"id":"fd1a61a4-f01c-4a03-a244-4149eff2fea8","arxiv_id":"1908.09040","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In exponential directed last-passage percolation, the full family of semi-infinite geodesics is characterized: unique coalescing trees in all but a countable dense set of directions, exactly two trees in the exceptional directions, which coincide with Busemann measure support and competition…","lead":"Researchers give the first complete description of all semi-infinite geodesics in directed last-passage percolation with exponential weights: in almost every direction there is a single coalescing family of optimal paths, and in a random countable dense set of exceptional directions there are exactly two, forming two coalescing trees.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central exponential characterization is secure; the only soft spot is the general-weight jump-process condition (3.5), which the paper explicitly leaves open for non-solvable weights and which does not affect Theorem 3.11.","rationale":"The Reader's weakest assumption is the same condition I would flag for the paper's broader claims: (3.5) is unproven for general weights. However, the paper's headline 'complete characterization' is specifically the exponential case, where (3.5) is a theorem from [22]. Thus the concern is scoped correctly by the authors and does not move the verdict. I also checked the internal dependencies: Theorem 3.2 and Proposition 7.1 give the support-coalescence dictionary; Theorem 3.5 correctly makes coalescence equivalent to (3.5); Theorem 3.7 and Lemma 3.6 give V_omega the competition-interface identification; and the exponential specialization verifies each hypothesis. The Palm-kernel calculation in Section 9 is a derivation from known queueing facts in Appendices C and D, not an independent assumption. No boundary or density issue in the measure-theoretic definition of supp mu_{x,y} appears to affect the results, since all uses are on compact subintervals or via the formal support definition (3.2). The one imprecision in Theorem 3.11(c) is minor. Verdict unchanged.","tokens_in":58358,"tokens_out":7156,"duration_ms":77164,"concrete_test":"Verify that [22, Theorem 3.4] applies verbatim to the Busemann process constructed in [27,36] and implies (3.5) for every pair x,y, not only adjacent edges, since Theorem 3.5 and Theorem 3.10(d) pass through this implication. Independently re-derive the last step of Theorem 3.11(c) with the added assumption xi in V_omega, or add a sentence treating the unique-geodesic case; if either check reveals a mismatch, the headline theorem still stands but the statement needs a qualifier.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reading the full manuscript, I cannot identify a load-bearing flaw in the central claim (Theorem 3.11). The complete characterization in the exponential model is assembled from: strict concavity and differentiability of the exponential shape (3.8), which gives (2.4); the jump-process condition (3.5), cited from [22, Theorem 3.4]; the no-nontrivial-bigeodesic condition (4.2), cited from [8,9]; and Coupier's at-most-two theorem (Theorem A.8). Each input is external and cited precisely, and the internal proofs connect them correctly: Theorem 3.10(d) supplies two coalescing sign-families in V_omega, and Theorem 3.11(a) then upgrades 'at least two' to 'exactly two' via Theorem A.8. The open status of (3.5) for general i.i.d. weights is explicitly recorded in Section 6, Problem 3, and the unconditional general-weight claims are carefully worded without it. The only blemish I found is in Theorem 3.11(c): for xi not in V_omega the split point s_xi(x) is infinity and the competition interface phi^{s_xi(x)} is not defined, although the intended convergence is already the unique-geodesic case of Theorem 3.10(c). This is a statement-level omission, not a threat to the main result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the full family of semi-infinite geodesics in planar directed last-passage percolation with i.i.d. continuous weights of finite p>2 moment. The authors introduce Busemann measures, the Lebesgue--Stieltjes measures associated with the monotone Busemann process, and use their supports to encode exceptional directions in which uniqueness or coalescence fails. The main general-weight results characterize the support of Busemann measures via disjoint geodesics (Theorem 3.1), relate coalescence points to isolated points of the support (Theorems 3.4 and 3.5), identify the random set V_omega of exceptional directions with competition-interface directions under the jump process condition (3.5) (Theorem 3.7), and describe uniqueness and coalescence of geodesics inside and outside V_omega (Theorems 3.9 and 3.10). In the exponential model, Theorem 3.11 gives a complete description: countably many dense exceptional directions in which every site emits exactly two semi-infinite geodesics forming left and right coalescing trees, and all other directions in which there is a unique coalescing tree of geodesics. Sections 4 and 5 develop the dynamical-systems interpretation through webs of instability and derive explicit statistics of instability points, including a connection to the zero set of simple symmetric random walk.","tokens_in":58500,"tokens_out":4369,"duration_ms":46484,"significance":"If the results are correct, Theorem 3.11 is the first complete characterization of semi-infinite geodesics in a KPZ-class lattice growth model, and the paper is likely to become a standard reference for the geometry of geodesics in exactly solvable last-passage percolation. The architecture of the proof is sound: the external inputs are cited precisely (Theorems A.1, A.4, A.7, and A.8; the jump-process condition (3.5) from [22]; the no-bigeodesic condition (4.2) from [8,9]), and the internal arguments correctly assemble these inputs. I particularly credit the careful separation of unconditional general-weight results from results that require condition (3.5), which the authors explicitly leave open for non-solvable weights in Section 6, Problem 3. The Palm-kernel arguments in Section 9 are handled in detail through Kallenberg's framework and lead to concrete, falsifiable statistics, including the unexpected comparison with the zero set of simple symmetric random walk.","major_comments":[],"minor_comments":[{"comment":"The statement quantifies over all xi in ri U, but for xi not in V_omega the split point s_xi(x) is infinity by definition (3.9), and the competition interface phi^{s_xi(x)} is not defined. Please either restrict the statement to xi in V_omega or add a convention for the case s_xi(x)=infinity, in which both alternative assumptions are vacuous and the conclusion should be the unique-geodesic convergence already contained in Theorem 3.10(c). This is a statement-level omission and does not affect the proof of the main characterization.","section":"Theorem 3.11(c)"},{"comment":"In the proof of part (c), the line 'call k = x dot hat e' appears to have a missing subscript: it should be k = x dot hat e_1. The same sentence should also specify that the convergence gamma_{x,v_n} -> gamma_{x,xi-} is in the sense of finite-segment convergence, consistent with the definition before the theorem.","section":"Proof of Theorem 3.11"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — read this one. The headline result, Theorem 3.11, is the first complete description of semi-infinite geodesics in a KPZ-class lattice model: a random countably infinite dense set V_omega of directions with exactly two geodesics from every site, left and right coalescing trees, and unique coalescing trees in all other directions. V_omega is identified both with competition-interface directions and with the discontinuities of the Busemann process. That is a substantial result, not an incremental step.\n\nWhat is actually new is the support-of-Busemann-measure framework (Theorems 3.1–3.9), the coalescence equivalences, the instability graphs, and the exponential-model statistics, including the SSRW zero-set connection for jump locations. The proofs are detailed, and the external inputs — Busemann process from [27,36], jump-process condition from [22], no-bigeodesic condition from [8,9], Coupier's at-most-two theorem — are cited precisely. The self-citations are not inflated; the cited results are independent proofs.\n\nThe soft spots are real but contained. Condition (3.5), that every support point of the Busemann measures is isolated, is open for general i.i.d. weights and the paper says so in Section 6. The most complete two-tree picture and the equality with competition-interface directions depend on that condition. It does not threaten Theorem 3.11, because the exponential case is proven. There is also a statement-level slip in Theorem 3.11(c): when xi is not in V_omega, s_xi(x) is infinity and the competition interface phi^{s_xi(x)} is undefined, so that part is not literally meaningful as written. The intended claim is just the unique-geodesic case of Theorem 3.10(c), and the fix is obvious. Minor blemish, not a load-bearing flaw.\n\nThe citation pattern is healthy. No code or data, but this is a proof paper and the proofs are the product. Anyone working in LPP, KPZ, or stochastic Hamilton-Jacobi theory should engage with this. It deserves a serious referee and should be accepted after minor revision. I would cite it and bring it to reading group.","headline":"First complete geometry of semi-infinite geodesics in exponential LPP, built on a genuinely new Busemann-measure framework; the central theorem holds up, with only minor statement-level blemishes.","tokens_in":59184,"tokens_out":4075,"would_cite":true,"duration_ms":37498,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35"],"pacs":[],"model":"deepseek-v4-flash","headline":"In exponential directed last-passage percolation, a random countably infinite dense set of exceptional directions carries exactly two semi-infinite geodesics per site—a left and a right one—each family forming a coalescing tree, while…","keywords":["last-passage percolation","semi-infinite geodesics","Busemann process","Busemann measures","coalescence","competition interface","KPZ universality","random Hamilton-Jacobi equations"],"falsifier":"To refute the general framework, find an i.i.d. weight distribution satisfying (1.1) with a realization in which some Busemann measure has a non-isolated support point—equivalently, a direction that is a limit of competition-interface directions without being one—since the jump process condition (3.5) and the equality $V_\\omega = \\{\\xi^*(T_x\\omega)\\}$ would then both fail; to refute the exponential statistics, check numerically whether the spacings of horizontal jump locations in a fixed exceptional direction follow the Catalan law $C_{n-1}2^{-(2n-1)}$ that matches the zero set of simple symmetric random walk.","tokens_in":58018,"feed_emoji":"🌳","tokens_out":11852,"duration_ms":98081,"temperature":0.7,"pith_summary":"This paper establishes the complete geometry of semi-infinite geodesics in the exponential planar directed last-passage percolation model: almost surely there is a random countably infinite dense set $V_\\omega$ of interior directions in which every lattice site is the root of exactly two geodesics with that asymptotic direction, a left one and a right one, with each of the two families forming a tree of coalescing paths. In every direction outside $V_\\omega$, each site has a unique geodesic and all such geodesics coalesce into a single tree. The exceptional set $V_\\omega$ is identified simultaneously with the set of asymptotic directions of competition interfaces and with the set of discontinuity directions of the Busemann process, so geodesic non-uniqueness coincides with non-continuity of Busemann functions. For general i.i.d. weights the paper introduces Busemann measures—Lebesgue–Stieltjes measures of the generalized Busemann functions indexed by direction—and shows that the support of these measures exactly records the directions where uniqueness or coalescence fails whenever a 'jump process' condition holds. The exactly solvable exponential case is the first KPZ-class lattice growth model for which the entire family of semi-infinite geodesics is completely accounted for.","feed_headline":"Exactly two geodesics per site in exceptional directions of LPP","feed_subtitle":"Outside a dense random set of directions, geodesics are unique and coalesce into one tree.","key_machinery":"The machinery is the Busemann measure: the signed Lebesgue–Stieltjes measure $\\mu_{x,y}$ obtained from the direction-indexed family of generalized Busemann functions $B^{\\xi\\pm}_{x,y}$, whose support records those directions in which the one-sided geodesics out of $x$ and $y$ fail to meet. The analysis runs through the support of these measures, the ordering and one-sided continuity of the Busemann geodesics, and two structural conditions: the regularity condition (2.4) on the shape function (strict concavity at non-differentiability points) and the jump process condition (3.5) (every point of every support is isolated). The jump process condition is what forces the exceptional set to be countable and to equal the competition-interface directions, and in the exponential model it holds by a theorem on the queueing representation of the Busemann functions; the strictly concave square-root shape function then gives $U_\\xi = \\{\\xi\\}$, so every geodesic is genuinely direction-directed.","core_discovery":"The paper's central discovery is that the set of exceptional directions of a realization is exactly the union $V_\\omega = \\bigcup_{x,y} \\operatorname{supp} \\mu_{x,y}$ of the supports of the Busemann measures, and that this set completely controls the geodesic geometry. In the exponential model, Theorem 3.10 and Theorem 3.11 prove that for every $\\xi \\notin V_\\omega$ the left and right Busemann geodesics $\\gamma_{x,\\xi-}$ and $\\gamma_{x,\\xi+}$ coincide for all $x$, form the unique $\\xi$-directed geodesic from each site, and all these geodesics coalesce; for every $\\xi \\in V_\\omega$, the two one-sided geodesics from each $x$ differ from the first step, never meet again, and are the only two $\\xi$-directed geodesics from $x$, while the left families (and separately the right families) coalesce into trees. The same theorems identify $V_\\omega$ with the set of asymptotic directions of competition interfaces and with the set of discontinuities of the Busemann process, establishing a three-way equivalence between geodesic non-uniqueness, interface directions, and jump discontinuities of the stochastic Hamilton–Jacobi solution.","pith_inferences":["If the jump process condition holds for general i.i.d. weights, as the paper's open problems anticipate, then the same dichotomy—one coalescing tree outside $V_\\omega$, two coalescing trees at each direction in $V_\\omega$—would govern geodesics in every planar directed LPP, making the exponential picture the universal one rather than an integrable special case.","The Busemann-measure framework uses only cocycle, monotonicity, and support properties, so it should transfer to undirected first-passage percolation and to stochastic Hamilton–Jacobi equations with a Hopf–Lax–Oleinik semigroup; in those models the support of the Busemann measures would serve as the full shock set, including branching shocks.","The random-walk statistics suggest a sharper universality prediction than the paper states: in any model where the jump condition holds, the law of the location of instability points on a line should be the same SSRW-zero-set law, which could be tested in non-integrable models by simulation."],"forward_implications":["For every site $x$ and every direction $\\xi \\notin V_\\omega$, the finite geodesic from $x$ to any sequence of endpoints with asymptotic direction $\\xi$ converges to the unique $\\xi$-directed semi-infinite geodesic; for $\\xi \\in V_\\omega$ it converges to the left or right geodesic according to the side of the competition interface on which the endpoints lie, so the limiting geometry of finite geode","The equality of $V_\\omega$ with the set of discontinuity directions of the Busemann process means that geodesic non-uniqueness is the same phenomenon as non-uniqueness of the backward Hamiltonian–Jacobi solution: the exceptional directions are exactly the shock directions of the discrete stochastic Burgers equation.","In the exponential model the Palm distribution of the locations of jumps of the Busemann process on a line is the zero set of simple symmetric random walk sampled at even times; consequently, the number of instability points in an $n\\times n$ box is almost surely $O(n^{3/2}\\sqrt{\\log n})$.","The set $V_\\omega$ is dense in the set of directions where the shape function is either non-differentiable or strictly concave, so the exceptional directions are not isolated artifacts—they occur throughout the entire interior of the direction space."],"supporting_citations":[{"why":"Proves the jump process condition (isolated supports of Busemann measures) for i.i.d. exponential weights, the key hypothesis behind the complete two-tree picture.","marker":"[22, Theorem 3.4]"},{"why":"Rules out three ξ-directed semi-infinite geodesics from a common vertex in the exponential model, which yields the 'exactly two' part of Theorem 3.11(a).","marker":"[16]"},{"why":"Supplies the Busemann geodesics' existence, directedness, coalescence, and the competition-interface characterization that the present paper extends to the full direction set.","marker":"[27]"},{"why":"Constructs the generalized Busemann process on an extended probability space without assumptions on the shape function, the object whose distribution functions become the Busemann measures.","marker":"[36]"},{"why":"Provides the strictly concave square-root shape function for exponential weights, which makes every interior direction a differentiability point and reduces U_xi to {xi}.","marker":"[47]"},{"why":"Supplies the inter-arrival law of the zero set of simple symmetric random walk sampled at even times, against which the instability-point statistics are matched.","marker":"[23]"}],"fun_headline_variants":["Exceptional LPP directions force exactly paired geodesics","LPP geodesic uniqueness fails exactly on a dense random set","Busemann supports give exact LPP geodesic splitting","Random walk emerges in LPP geodesic instability statistics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the support of every Busemann measure consists only of isolated points (the jump process condition (3.5)), which is proven for exponential weights but open for general i.i.d. weights; if it fails, the equalities between the exceptional-direction set, the competition-interface directions, and the discontinuity set of the Busemann process, and with them the full two-tree classification, would break.","fun_headline_variants_meta":{"raw":{"variants":["Exceptional LPP directions force exactly paired geodesics","LPP geodesic uniqueness fails exactly on a dense random set","Busemann supports give exact LPP geodesic splitting","Random walk emerges in LPP geodesic instability statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001174,"raw_usage":{"total_tokens":4847,"prompt_tokens":933,"completion_tokens":3914,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":3849}},"tokens_in":549,"tokens_out":3914,"duration_ms":28366,"temperature":1.0,"reasoning_tokens":3849,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:23:54.975530+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To refute the general framework, find an i.i.d. weight distribution satisfying (1.1) with a realization in which some Busemann measure has a non-isolated support point—equivalently, a direction that is a limit of competition-interface directions without being one—since the jump process condition (3.5) and the equality $V_\\omega = \\{\\xi^*(T_x\\omega)\\}$ would then both fail; to refute the exponential statistics, check numerically whether the spacings of horizontal jump locations in a fixed exceptional direction follow the Catalan law $C_{n-1}2^{-(2n-1)}$ that matches the zero set of simple symmetric random walk.","supporting_citations":[{"cited_title":"Multiple geodesics with the same direction","cited_arxiv_id":null,"evidence_quote":"Rules out three ξ-directed semi-infinite geodesics from a common vertex in the exponential model, which yields the 'exactly two' part of Theorem 3.11(a)."},{"cited_title":"Geodesics and the competition interface for the corner growth model","cited_arxiv_id":null,"evidence_quote":"Supplies the Busemann geodesics' existence, directedness, coalescence, and the competition-interface characterization that the present paper extends to the full direction set."},{"cited_title":"Busemann functions and Gibbs measures in directed polymer models on Z2","cited_arxiv_id":null,"evidence_quote":"Constructs the generalized Busemann process on an extended probability space without assumptions on the shape function, the object whose distribution functions become the Busemann measures."},{"cited_title":"Nonequilibrium behaviour of a many particle process: density profile and local equilibria","cited_arxiv_id":null,"evidence_quote":"Provides the strictly concave square-root shape function for exponential weights, which makes every interior direction a differentiability point and reduces U_xi to {xi}."},{"cited_title":"An introduction to probability theory and its applications","cited_arxiv_id":null,"evidence_quote":"Supplies the inter-arrival law of the zero set of simple symmetric random walk sampled at even times, against which the instability-point statistics are matched."}],"review_version":1}