{"id":"d5e41ac9-19e2-447c-ba01-59e5a2f0fc41","arxiv_id":"2502.00942","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For general i.i.d. weights, the midpoint of the geodesic between (0,0) and (n,n) lying at position n/2+tn has probability e^{-2nJ_t(μ0)+o(n)}.","lead":"This paper proves a large deviation formula for the probability that the optimal path in a random landscape strays far from the diagonal. The result links geodesic fluctuations in last-passage percolation to the known large deviation rate of passage times, and it confirms an open conjecture for exponential weights.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3 upper bound: the displayed estimate for (3.7) uses a squared probability that cannot follow from the event being bounded, so the proof of Theorem 1.1 is incomplete as written.","rationale":"The reader's weakest_assumption was Proposition 2.5, the Gaussian left-tail estimate. That is a legitimate secondary concern, and the paper's own sketch is not a full proof for the general weight class. However, the more direct obstruction to Theorem 1.1 is the derivation of the upper bound in Section 3. Even if Proposition 2.5 is granted, the displayed estimate for (3.7) contains a squared probability that is not a consequence of the event being bounded: the independent subevent in which both summands exceed the threshold has probability of order e^{-2nJ}, which already exceeds the printed right-hand side of order e^{-4nJ}. This is not a mere notation issue; it is the exact step that produces the rate 2J_t(mu0) in the upper bound. The result may well be true, since a corrected argument using the large deviation of the sum of the two independent passage times should give the same exponent, but that argument is absent. Proposition 2.5 should still be checked, but it is not the single most load-bearing point. Because the concern is a proof gap rather than evidence that the theorem is false, the appropriate verdict remains conditional, matching the reader's assessment.","tokens_in":13424,"tokens_out":21306,"duration_ms":216162,"concrete_test":"Check the printed chain for (3.7) by verifying the set inclusion {A>=m} cap {B>=m} subset {A+B>=2m}, where A and B are independent copies of G_{0,(floor(n/2+tn),floor(n/2-tn))}. For any weights satisfying (1.2), the left-hand subevent has probability at least e^{-2n(J_t(mu0-delta)+o(1))}, while the printed upper bound P(G_{0,q} >= 2m)^2 has order e^{-4n(J_t(mu0-delta)+o(1))}; hence the displayed squared inequality is false for large n. Then re-derive Section 3 with the corrected unsquared bound P(G_{0,q} >= 2m) and verify that this alone yields the desired e^{-2n(J_t(mu0)-epsilon)} upper bound without the square.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central upper bound in Theorem 1.1 rests on the chain from (3.4) to (3.7). After conditioning on Mid_{0,n}=p=(n/2+tn,n/2-tn), the paper needs to bound P(G_{0,p}+G_{p,(n,n)} >= 2mu0 n - 2delta n). In (3.7) this is rewritten with G_{p,(n+2tn,n-2tn)}; if the justification is that the two second-segment passage times have the same law under reflection/translation, that should be stated, since the displayed equality is otherwise not transparent. More seriously, the next displayed inequality is P(G_{0,p}+G_{p,(n+2tn,n-2tn)} >= 2m) <= P(G_{0,q} >= 2m)^2, with m=(mu0-delta)n and q=(n+2tn,n-2tn). This is not a valid consequence. Let A=G_{0,p} and B=G_{p,q}; they are independent and have the same marginal tail rate J_t. The event {A>=m} cap {B>=m} is contained in {A+B>=2m} and has probability about e^{-2n J_t(m)}, while the printed right-hand side is of order e^{-4n J_t(m)}. Thus the printed inequality is false as written, and the claimed e^{-2n(J_t(mu0)-epsilon)} bound is not obtained. The natural fix is to bound the sum event directly by P(G_{0,q} >= 2m) or by a legitimate independent-sum large-deviation estimate, but that argument is not supplied in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies linear-scale transversal fluctuations of the geodesic midpoint in planar last-passage percolation with general i.i.d. weights satisfying a mild exponential-moment assumption. The main result, Theorem 1.1, asserts that for directions t with mu_0 > mu_t, both P(Mid_{0,n} . e1 >= n/2 + tn) and P(Mid_{0,n} . e1 = n/2 + tn) decay as exp(-2n J_t(mu_0) + o(n)), where J_t is the right-tail large-deviation rate function of the point-to-point passage time and mu_t is the shape function. The upper bound is proved by decomposing the midpoint event according to the total passage time and estimating a sum of two independent passage times; the lower bound is proved by planting a high-weight path and showing the geodesic stays near it. As an application, the paper derives the exponential-LPP corner-path probability (4/e^2)^{n+o(n)}, verifying a conjecture of Liu. The central upper-bound estimate contains an invalid inequality, the left-tail estimate on which it relies is only sketched, and the lower-bound section has a circular ordering in its averaging argument.","tokens_in":13744,"tokens_out":21752,"duration_ms":210495,"significance":"If the proof gap is repaired, the result would be a significant contribution: it would give the first large-deviation principle for geodesic midpoint fluctuations in a general non-integrable LPP setting, express the rate function in terms of the known passage-time rate function J_t and the shape function, and verify Liu's conjecture for exponential LPP. The paper is clearly written and uses standard tools; it introduces no fitted parameters or ad hoc objects. The lower-bound construction via planted paths is conceptually appealing. However, the current manuscript does not establish the upper bound because of the erroneous inequality in Section 3, and Proposition 2.5 requires a complete proof; these are central, not cosmetic, issues.","major_comments":[{"comment":"The displayed inequality P(G_{0,p} + G_{p,(n+2tn,n-2tn)} >= 2m) <= P(G_{0,(n+2tn,n-2tn)} >= 2m)^2 is not valid. Writing A = G_{0,p} and B = G_{p,(n+2tn,n-2tn)}, the event {A + B >= 2m} is not contained in {A >= m} and {B >= m}; one variable can be less than m and the other sufficiently larger. For independent A and B with common upper-tail rate J_t, the probability {A + B >= 2m} is typically of order e^{-2n J_t(mu_0 - delta)}, whereas the printed square is of order e^{-4n J_t(mu_0 - delta)}. Thus the displayed bound is false and the upper bound in Theorem 1.1 does not follow as written. A correct proof needs a legitimate independent-sum large-deviation upper bound, for example via exponential Chebyshev optimized over theta, using the convexity of J_t; this argument is not supplied. This is the load-bearing gap in the paper.","section":"Section 3, displayed estimate following (3.7)"},{"comment":"The Gaussian left-tail estimate P(G_{0,(n/2,n/2)} <= mu_0 n - epsilon n) <= e^{-c epsilon^2 n^2} is load-bearing for the upper bound through its use at (3.6). The proof given, however, is only a sketch. In particular, the key estimate P(G^K_{0,(n/2,n/2)} <= (mu_0 - 3epsilon/4)n) <= e^{-cn} is cited to [7, Lemma 2.2] without a statement, and the adaptation from [7, Section 4.1] is noted in the text as not formally stated. The sketch also does not fully justify the independence of the strip-restricted passage times G^K across i, nor the precise relation between the parameters delta, K, and epsilon. Since this estimate is used to absorb the first term in the upper-bound decomposition, the paper should state it as a theorem or proposition with a complete proof, or give a precise reference that covers the general weight class (1.2).","section":"Section 2, Proposition 2.5"},{"comment":"The averaging argument in the first part of Section 4 is circular as written. The chain leading to P(E_0) >= (1/(2n^2)) e^{-2n(J_t(mu_0)+3epsilon)} uses the lower bound P(H_0) >= e^{-2n(J_t(mu_0)+epsilon)}, which is exactly the weak lower bound (4.8) that is only proved later in Sections 4.1 and 4.2. The argument should be reordered: prove (4.8) first, and then use the translation-invariance and path-monotonicity argument to upgrade the '>=' lower bound to the exact equality lower bound in Theorem 1.1. The mathematical content is repairable, but the current logical order makes the proof unverifiable as written.","section":"Section 4, equations (4.9)-(4.10)"}],"minor_comments":[{"comment":"The equality replacing G_{p,(n,n)} by G_{p,(n+2tn,n-2tn)} is not an identity of random variables; at best it is an equality in distribution, using the reflection and translation symmetries of the i.i.d. weight field. This should be stated explicitly.","section":"Section 3, first display after (3.7)"},{"comment":"The notation involving '/BD' and the summation limits (for example, '2n sum' and 'n^2-1 sum') appears garbled and should be cleaned up so that the indices and ranges are unambiguous.","section":"Section 4, notation"},{"comment":"The proof of Proposition 2.3 uses the symmetry J_{-t}(r) = J_t(r) without stating it; this follows from the reflection symmetry of the i.i.d. model and should be noted.","section":"Section 2, Proposition 2.3"},{"comment":"The paper repeatedly uses expressions such as n/2 + tn where n/2 + tn is not an integer; the floor notation in Theorem 1.1 is introduced, but the proofs should consistently verify that the displayed equalities and inequalities are unaffected by the rounding.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely salvageable: the main theorem is plausible and the lower-bound construction is substantial, but the invalid inequality in Section 3 is a central gap and the left-tail estimate in Proposition 2.5 needs a complete proof. I recommend major revision rather than rejection, and I would ask the authors to supply the missing independent-sum large-deviation argument and a full proof of Proposition 2.5 before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper has the right theorem and the right reduction, but the upper-bound proof in Section 3 is not correct as written. The displayed line after (3.7) bounds the sum of two independent passage times by the square of a single passage-time tail. That inequality goes the wrong way, and it is load-bearing: without it the exponent 2J_t(mu0) in Theorem 1.1 is not obtained.\n\nWhat is genuinely new: an LDP for midpoint transversal fluctuations in LPP with general i.i.d. weights under mild exponential moment assumptions, expressed through the passage-time rate function J_t. The reduction to J_t is clean, and the lower-bound construction (planting the path) is a sensible adaptation of Agarwal–Basu [2]. The corollary for exponential LPP, giving the corner-path probability (4/e^2)^{n+o(n)}, confirms Liu's conjecture and is a nice payoff. The paper is honest about what it borrows: it credits [2] for the leading-order constant in the exponential case and for the geometric ideas.\n\nThe soft spot is the upper bound. From (3.4) to (3.7), the paper needs to control P(G_{0,p}+G_{p,(n,n)} >= 2(mu0-delta)n). The next displayed equality replaces the second term with G_{p,(n+2tn,n-2tn)}; that is not explained and the endpoint looks wrong. Then the inequality asserts this probability is at most P(G_{0,(n+2tn,n-2tn)} >= 2(mu0-delta)n)^2. For independent A,B with the same tail rate, P(A+B >= 2m) is not bounded above by P(A >= m)^2; the event {A>=m, B>=m} is a subset of the sum event, so the product is a lower bound, not an upper bound. A union bound would give a single exponential e^{-n(J_t - epsilon)}, too weak by a factor of two in the exponent. The argument as printed does not prove Theorem 1.1, and the fix needs a real two-scale or concentration argument, not a typo correction.\n\nI also share the reader's mild concern about Proposition 2.5: it is cited from Kesten and sketched, but it is doing real work in (3.6). I would want it stated as a proposition with a full proof or a precise reference.\n\nIf the upper bound can be repaired, this is a solid contribution worth publishing. As it stands, the result is plausible but the proof is incomplete. My recommendation: send to a serious referee, but with the explicit expectation that Section 3 must be fixed. This should not be accepted as is.","headline":"The general-weight midpoint LDP is a real contribution, but Section 3's upper bound has a false inequality that leaves Theorem 1.1 unproved as written.","tokens_in":14302,"tokens_out":5736,"would_cite":false,"duration_ms":51387,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60K37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a large deviation principle for the geodesic midpoint in general last-passage percolation, with rate function $2J_t(\\mu_0)$, and uses it to verify the exponential corner-path conjecture from [18].","keywords":["last-passage percolation","large deviations","geodesic midpoint","transversal fluctuations","KPZ universality class","shape function","corner path","exponential weights"],"falsifier":"Using the exact contour-integral formula in [18], compute $P((n,0)\\in \\Gamma_{0,n})$ for rate-one exponential weights at increasing $n$; Theorem 1.1 and Corollary 1.3 predict $-(1/n)\\log P\\to 2-2\\log 2$, so a numerical sequence converging elsewhere would refute the claimed rate. Equally decisive: exhibit an i.i.d. weight distribution satisfying (1.2) for which $P(G_{0,(n/2,n/2)}\\le \\mu_0 n-\\epsilon n)$ decays slower than $e^{-c\\epsilon^2 n^2}$; that would break the upper-bound proof as written.","tokens_in":13223,"feed_emoji":"🧮","tokens_out":13779,"duration_ms":118866,"temperature":0.7,"pith_summary":"This paper establishes the large deviation rate for transversal fluctuation of the midpoint of the geodesic in last-passage percolation with general i.i.d. weights, going beyond exactly solvable models. The main theorem says that for each direction $t$ with $\\mu_0 > \\mu_t$, the probability that the midpoint of the geodesic from $(0,0)$ to $(n,n)$ lands at horizontal displacement $\\lfloor n/2+tn\\rfloor$ decays as $e^{-2n(J_t(\\mu_0)+o(1))}$, where $J_t$ is the right-tail large deviation rate function of the last-passage value and $\\mu_0$ is the diagonal shape constant. This ties the random geometry of the optimal path to the known large deviations of passage times, making geodesic large deviations accessible for general weight distributions. As a concrete consequence, for i.i.d. rate-one exponential weights the result confirms the conjecture communicated with [18]: the geodesic follows the corner path $(0,0)\\to(n,0)\\to(n,n)$ with probability $(4/e^2)^{n+o(n)}$.","feed_headline":"A single rate function governs geodesic midpoint detours","feed_subtitle":"The rate is set by passage-time large deviations, and the exponential corner-path conjecture is confirmed.","key_machinery":"The central object is the right-tail rate function $J_t(r)=-\\lim_{n\\to\\infty} n^{-1}\\log P\\bigl(G_{0,(\\lfloor n/2+tn\\rfloor,\\lfloor n/2-tn\\rfloor)}\\ge rn\\bigr)$ and the shape function $\\mu_t$. The midpoint event is rewritten using the exact identity\n$$\\{Mid_{0,n}=(n/2+tn,n/2-tn)\\} = \\{G_{0,(n/2+tn,n/2-tn)}+G_{(n/2+tn,n/2-tn),(n,n)}=G_{0,(n,n)}\\}.$$\nFor the upper bound, the paper splits on the event $G_{0,(n,n)}\\le 2\\mu_0 n-2\\delta n$, controlled by a Gaussian lower-tail estimate (Proposition 2.5, cited from [17] and adapted from [7]), and uses the right-tail large deviation bound (Proposition 2.4) on the complementary event. For the lower bound, the paper plants a sequence of high-weight geodesic segments between points spaced $2\\delta^5 n$ apart, concatenates them, and uses a coalescence argument together with the BKR inequality to show the true geodesic must pass within $O(\\delta n)$ of the planted midpoint. The case $t=1/2$ uses a separate corner-path planting argument.","core_discovery":"Under the assumptions (1.2)---finite exponential moment, continuous weights, unbounded support---the paper proves Theorem 1.1: for any fixed $0<t\\le 1/2$ with $\\mu_0>\\mu_t$ and any $\\epsilon>0$, for all large $n$,\n$$$e^{{-2n(J_t(\\mu_0)+\\epsilon)}}$ \\le P\\bigl(Mid_{0,n}\\cdot e_1 = \\lfloor n/2+tn\\rfloor\\bigr) \\le P\\bigl(Mid_{0,n}\\cdot e_1 \\ge \\lfloor n/2+tn\\rfloor\\bigr) \\le $e^{{-2n(J_t(\\mu_0)-\\epsilon)}}$.$$\nThus the midpoint transversal fluctuation obeys a large deviation principle at speed $n$ with rate function $2J_t(\\mu_0)$. The companion Proposition 1.2 gives the point-to-line endpoint the same statement with rate $J_t(\\mu_0)$. For rate-one exponential weights, Corollary 1.3 follows immediately: $J_{1/2}(x)=x-\\log x-1$ and $\\mu_0=2$, so the probability that the geodesic from $(0,0)$ to $(n,n)$ contains the corner point $(n,0)$ is $(4/e^2)^{n+o(n)}$, verifying the conjecture in [18].","pith_inferences":["A direct asymptotic evaluation of the exact contour-integral formula in [18] at $t=1/2$ would give an independent check of the rate $2-2\\log 2$ without relying on Theorem 1.1; the paper does not perform that evaluation.","The lower-bound planting argument is flexible enough that the same rate function likely governs the probability that the entire geodesic stays far from the diagonal, not just the midpoint; the paper proves only the midpoint statement.","If the conjectured inequality $2J_t(\\mu_0)\\le$ the corresponding rate for a uniformly chosen path holds for general weights, it would yield new constraints on the shape function $\\mu_0$; testing both sides in exponential LPP is a finite calculation.","The proof hinges on a Gaussian lower-tail estimate that is only sketched; finding a weight distribution satisfying (1.2) with a weaker lower tail would not disprove the theorem but would force a different upper-bound argument."],"forward_implications":["For exponential weights, the probability that the geodesic from $(0,0)$ to $(n,n)$ uses the corner path $(0,0)\\to(n,0)\\to(n,n)$ is $(4/e^2)^{n+o(n)}$, larger by an exponential factor than the $2^{-2n+o(n)}$ rate for a uniformly chosen up-right path.","For every direction $t$ with $\\mu_0>\\mu_t$, the large deviation rate of the midpoint is exactly $2J_t(\\mu_0)$; the right-tail passage-time rate function therefore determines this geometric large deviation in full.","The point-to-line geodesic endpoint fluctuates with rate $J_t(\\mu_0)$, exactly half the midpoint rate, giving a clean comparison between point-to-point and point-to-line geometry.","Because the proof requires only finite exponential moments and mild regularity, the same large deviation rate holds for all weight distributions in the class (1.2), not just solvable models."],"supporting_citations":[{"why":"Supplies the Gaussian lower-tail estimate for passage times (Proposition 2.5) that anchors the upper-bound proof.","marker":"[17]"},{"why":"Gives the adaptation of the lower-tail estimate and the delocalization argument the paper's sketch follows.","marker":"[7]"},{"why":"Contains the conjecture that the exponential corner-path probability is $(4/e^2)^{n+o(n)}$, verified as Corollary 1.3.","marker":"[18]"},{"why":"Contributes the planted-path and coalescence-geometry construction used in the lower-bound proof.","marker":"[2]"},{"why":"Establishes existence, convexity, and continuity of the right-tail rate function $J_t(r)$ used throughout.","marker":"[14]"},{"why":"Gives continuity of the shape function $\\mu_t$, justifying that $\\mu_0>\\mu_t$ holds on a nontrivial interval.","marker":"[19]"},{"why":"Provides the value $\\mu_0=2$ for rate-one exponential LPP, needed for Corollary 1.3.","marker":"[21]"},{"why":"Supplies the BKR inequality used to control disjoint occurrences in the lower-bound argument.","marker":"[4]"}],"fun_headline_variants":["Geodesic midpoint fluctuations obey a large deviation principle","Corner-path conjecture proved: probability (4/e^2)^(n+o(n))","Midpoint detour rate function tied to passage-time deviations","Exponential LPP confirms corner-path asymptotic","Large deviations for geodesic midpoint at speed n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the diagonal passage time has a much thinner lower tail than an exponential one: $P(G_{0,(n/2,n/2)}\\le \\mu_0 n-\\epsilon n)\\le e^{-c\\epsilon^2 n^2}$, an estimate the paper cites from [17] and adapts from [7] but only sketches for its weight class; if that estimate fails for some weights satisfying (1.2), the upper bound in Theorem 1.1 would not be established.","fun_headline_variants_meta":{"raw":{"variants":["Geodesic midpoint fluctuations obey a large deviation principle","Corner-path conjecture proved: probability (4/e^2)^(n+o(n))","Midpoint detour rate function tied to passage-time deviations","Exponential LPP confirms corner-path asymptotic","Large deviations for geodesic midpoint at speed n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000376,"raw_usage":{"total_tokens":2028,"prompt_tokens":993,"completion_tokens":1035,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":952}},"tokens_in":609,"tokens_out":1035,"duration_ms":9193,"temperature":1.0,"reasoning_tokens":952,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:10:18.026211+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using the exact contour-integral formula in [18], compute $P((n,0)\\in \\Gamma_{0,n})$ for rate-one exponential weights at increasing $n$; Theorem 1.1 and Corollary 1.3 predict $-(1/n)\\log P\\to 2-2\\log 2$, so a numerical sequence converging elsewhere would refute the claimed rate. Equally decisive: exhibit an i.i.d. weight distribution satisfying (1.2) for which $P(G_{0,(n/2,n/2)}\\le \\mu_0 n-\\epsilon n)$ decays slower than $e^{-c\\epsilon^2 n^2}$; that would break the upper-bound proof as written.","supporting_citations":[{"cited_title":"1180, Springer, Berl in, 1986, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian lower-tail estimate for passage times (Proposition 2.5) that anchors the upper-bound proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the adaptation of the lower-tail estimate and the delocalization argument the paper's sketch follows."},{"cited_title":"Theory Related Fields 184 (2022), no","cited_arxiv_id":null,"evidence_quote":"Contains the conjecture that the exponential corner-path probability is $(4/e^2)^{n+o(n)}$, verified as Corollary 1.3."},{"cited_title":"Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation","cited_arxiv_id":"2405.18056","evidence_quote":"Contributes the planted-path and coalescence-geometry construction used in the lower-bound proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes existence, convexity, and continuity of the right-tail rate function $J_t(r)$ used throughout."},{"cited_title":"Martin, Limiting shape for directed percolation models , Ann","cited_arxiv_id":null,"evidence_quote":"Gives continuity of the shape function $\\mu_t$, justifying that $\\mu_0>\\mu_t$ holds on a nontrivial interval."},{"cited_title":"Rost, Nonequilibrium behaviour of a many particle process: densi ty profile and local equi- libria, Z","cited_arxiv_id":null,"evidence_quote":"Provides the value $\\mu_0=2$ for rate-one exponential LPP, needed for Corollary 1.3."},{"cited_title":"Hales, The van den Berg–Kesten-Reimer operator and inequality for infinite spaces , Bernoulli 24 (2018), no","cited_arxiv_id":null,"evidence_quote":"Supplies the BKR inequality used to control disjoint occurrences in the lower-bound argument."}],"review_version":1}