{"id":"76d7f589-131e-4ffe-a438-4b795451a752","arxiv_id":"2608.11378","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Corner paths are nearly the most likely geodesics in last passage percolation: they beat all but an exponentially small fraction of paths, and in exponential LPP their probability matches the maximum up to a sharp asymptotic constant under a stated moderate deviation assumption.","lead":"This paper studies which path shapes are most likely to be the geodesic, the highest-weight directed path, in last passage percolation. It proves that corner hugging paths beat all but a vanishing fraction of competitors, and it pins down the asymptotic probability in the exponential case up to one unproved moderate deviation estimate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3's unproved moderate-deviation range is the load-bearing gap: without c_l = 1/48 for t up to n^{1/6+ζ}, the sharp √n coefficient in Theorem 2 is not established.","rationale":"The reader's weakest_assumption identifies exactly the argument's keystone. The upper bound in Theorem 2 is unconditional and appears sound; the lower bound is the only place where the √(2n) coefficient can arise. In that proof, Lemma 4 converts Assumption 3 into a lower bound for P(A3) at t ≈ α n^{1/6}, and the optimization over α is forced by the balance between the linear gain and cubic penalty; a shorter-range estimate cannot deliver the constant 4/3. The paper states this limitation explicitly and does not overclaim: it marks Assumption 3 as an assumption rather than a proved theorem. Therefore CONDITIONAL is the appropriate verdict, and my stress-test does not change it. Secondary proof gaps flagged in the text, such as the omitted off-maximum terms in Corollary 2 and the non-diagonal uniformity note in Lemma 4, are genuine but subordinate: they would not affect the central conditional statement if Assumption 3 is supplied.","tokens_in":29570,"tokens_out":24887,"duration_ms":209307,"concrete_test":"Independently re-derive the point-to-line lower-tail estimate needed in Lemma 4 from the beta-ensemble/steepest-descent method of [Bas+25, Theorem 1.6], keeping explicit track of the allowed range of t. The decisive check is whether the t = O(n^{1/10}) restriction is an intrinsic error-bar barrier or an artifact of the proof technique; if the argument extends to t = c n^{1/6} for fixed c ∈ [T, ρ], then Assumption 3 is verified and the equality in Theorem 2 follows. If the extension fails, compute the best available c_l(t) in the required range and compare the resulting liminf with 4/3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is Assumption 3 (Section 2): the sharp lower-tail moderate deviation bound log P(L(0,0;n) ≤ 2µn − t n^{1/3}) ≥ −(1/48)t^3 + o(t^3) is required uniformly for T ≤ t ≤ ρ n^{1/6+ζ}. It enters the proof through Lemma 4 and is used in the lower bound of Theorem 2 at the single scale t = α n^{1/6} with α = (6c_l)^{−1/2}; the displayed optimization forces this scale. The paper notes that the available result [Bas+25, Theorem 1.6] only covers t = O(n^{1/10}), so the required range is unproved. Without Assumption 3, the unconditional lower bound in Theorem 2 is only liminf ≥ 1/(3√(3c_l)) with c_l an unspecified constant from [Bas+21, Theorem 2]; the claimed sharp equality in Theorem 2, expression (5), and the √n-order modality of corner paths therefore rest on an open estimate. This is an honestly flagged limitation rather than an internal inconsistency; Corollary 2 has an additional unproved half-space range, but the central claim depends on Assumption 3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the probability that a prescribed path is the geodesic in last passage percolation on Z^d, with emphasis on the planar exponential case. The authors introduce three mechanisms favoring 'corner' paths and prove: (i) unconditionally, under subcritical Assumptions 1 and 2, any path with linearly many corners is exponentially less likely to be the geodesic than the corner path (Theorem 1 and Corollary 1); (ii) in exponential LPP, an unconditional upper bound limsup ≤ 4/3 for the √n correction to the modal probability, and a lower bound of 1/(3√(3c_l)) in terms of an unspecified constant c_l, with equality to 4/3 under an explicit Assumption 3 on the sharp lower-tail moderate deviation rate for point-to-line times (Theorem 2); (iii) a conditional comparison showing corner paths beat staircase paths with k≥8 segments (Corollary 2); and (iv) a monotonicity theorem for Bernoulli weights with small p under bump and unwind moves (Theorem 3). The paper is transparent about the unproved nature of Assumption 3.","tokens_in":29897,"tokens_out":10484,"duration_ms":86837,"significance":"If the sharp results were unconditional, the paper would make a substantial contribution: it gives the first asymptotically sharp two-term expansion for the modal geodesic probability in exponential LPP, with a parameter-free constant 4/3 arising from optimizing competing exponents, and it rigorously establishes a corner-path optimality principle in several settings. The unconditional Theorem 1 and Corollary 1 are solid and of independent interest, and Theorem 3 provides a complete small-p monotonicity picture. The honest flagging of Assumption 3 is a strength, but it means the headline sharp equality is conditional on an open moderate-deviation estimate. The paper does not rely on self-citation or fitted parameters; the constant comes from the structure of the moderate deviation exponents.","major_comments":[{"comment":"The sharp equality in Theorem 2, the displayed expression (5), and the claim that corner paths are modal up to √n order all rest on Assumption 3, which requires the log-probability lower bound −(1/48)t^3 + o(t^3) uniformly for T ≤ t ≤ ρ n^{1/6+ζ}. As the paper notes, the best available result [Bas+25, Theorem 1.6] covers only t = O(n^{1/10}), so the required range is not proved. Without Assumption 3, the unconditional lower bound in Theorem 2 is only liminf ≥ 1/(3√(3c_l)) with c_l an unspecified constant from [Bas+21, Theorem 2]. The authors should either prove the needed moderate-deviation range or explicitly present Theorem 2 and (5) as conditional on an open conjecture and discuss the status of that conjecture in more detail.","section":"Section 2, Theorem 2 and Assumption 3"},{"comment":"The proof of Lemma 4 transfers the diagonal point-to-line moderate deviation lower bound to the maximum over starting points (0,k) with k ≤ a = ⌊n^{2/3−ε}⌋ by using a lower-tail penalty for the non-diagonal point-to-point passage times X_k over distance b = n^{1−ε}. The text states that one should prove uniformity across directions and that the bound from [LR10, Theorem 2] suffices, but no proof or precise statement of the required non-diagonal bound is supplied. Since Lemma 4 is used in the lower-bound part of Theorem 2 and in the derivation of the 4/3 constant under Assumption 3, this gap must be closed or explicitly referenced.","section":"Section 4, Lemma 4"},{"comment":"The proof of Corollary 2 contains two explicit omissions: the off-maximum terms in the union bound are not bounded ('We do not show carefully how to bound the off-maximum terms for reasons of length'), and the argument assumes a sharp half-space moderate deviation constant c_h = 1/384 over a wide but unspecified range of t. As stated, Corollary 2 is therefore not proved. The corollary should be restated as conditional on the same assumptions used in its proof, or the missing estimates must be supplied.","section":"Section 4, Corollary 2"}],"minor_comments":[{"comment":"The uniform moderate deviation estimate (3) is quoted for T ≤ t ≤ ρ n^{2/3}, while the proof uses it only for t ≤ ρ n^{1/2}; the reference to [Led18] is described as 'an indication of the proof', so a precise statement of the proven range and a complete citation for exponential weights would improve the paper.","section":"Section 4, proof of the upper bound in Theorem 2"},{"comment":"The statement says 'take k ≥ K, where K is a deterministic constant' but the value K is not specified; the proof indicates k ≥ 8 under the auxiliary assumptions on c_l and c_h. The statement should either specify the assumptions or the constant.","section":"Section 4, Corollary 2 statement"},{"comment":"The caption says the right panel is for n = 30, but the displayed matrix appears to be 8×8; please check the figure and caption.","section":"Section 1, Figure 2 caption"},{"comment":"The notation p_0 in the displayed bound is used both as the limit of p_α as α→0 and as a fixed constant p_0 > 0 in Lemma 3 of Section 4; this clash of notation could confuse readers and should be resolved.","section":"Section 3, proof of Theorem 1"},{"comment":"The proof says 'The case α = 1 has been treated in Appendix B', but Appendix B concerns the corner-path probability, not the transversal fluctuation bound; please add the missing reference or explanation.","section":"Appendix C, Proposition 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its main conditional input (Assumption 3), and the unconditional results are solid. However, the sharp 4/3 coefficient is the headline and it depends on an open moderate-deviation range; combined with the explicitly omitted off-maximum terms in Corollary 2 and the unproved non-diagonal uniformity in Lemma 4, this makes the manuscript not yet suitable for acceptance. I would encourage a revision that either proves or precisely cites the missing estimates, or reframes the affected statements as conditional results with the gaps clearly stated in the theorem statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead McKeown–Nakajima on most likely geodesics in LPP. Short version: it earns a serious referee. Theorem 1 — exponential separation for paths with linearly many corners in general dimension — and the unconditional upper bound in Theorem 2 are real progress. Both are worth having. The claimed sharp √n coefficient, however, rests on Assumption 3, an unproved moderate-deviation range for point-to-line passage times. The authors say this openly, and the stress-test note is right: without c_l = 1/48 up to t ~ n^{1/6+ζ}, the equality in Theorem 2 and expression (5) are not established.\n\nWhat the paper does well: it identifies three concrete mechanisms favoring corner paths, and the proof strategies — flip competition for Theorem 1, tilt/gain optimization for Theorem 2, discrete Green's theorem for Theorem 3 — are natural and mostly convincing. The Bernoulli monotonicity theorem is a nice combinatorial result with an explicit polynomial p_n lower bound, and the paper honestly notes it does not get a uniform p_n. There are no fitted parameters; the constant 4/3 comes from optimizing exponents in external moderate-deviation inputs. The citation pattern is honest, crediting [AC21] and [Alb+25] properly.\n\nSoft spots, in proportion:\n\n- Assumption 3 is load-bearing. The best available result [Bas+25] covers only t = O(n^{1/10}), while the proof needs t up to n^{1/6+ζ}. The unconditional lower bound has an unspecified c_l, so the sharp statement is conditional. This is handled honestly, but the preprint should say in the abstract that the sharp coefficient is conditional on an open estimate.\n\n- Lemma 4 applies the diagonal moderate-deviation estimate (3) to non-diagonal point-to-line times. The text says [LR10, Theorem 2] suffices because only a constant is needed, but no derivation is supplied. Needs a real argument or a precise reference.\n\n- Corollary 2's proof explicitly omits bounding the off-maximum terms, saying they can be handled like (10) with more factors. Plausible, but not written. It also assumes a half-space moderate-deviation range beyond what is proved. Minor to moderate gap.\n\n- In Appendix C, the proof of Proposition 4 says “We may estimate P(I^c) using Proposition 4” — self-referential; should be Proposition 3. A typo, but a glaring one in a proof about independence.\n\nSummary: this paper is for researchers in integrable probability and LPP. The unconditional results and the framework are the main value; the sharp constant under Assumption 3 is a well-posed open problem that the paper correctly identifies. I would cite Theorem 1 and the upper bound of Theorem 2, and I would treat the conditional result with its caveat firmly in mind.\n\nRecommendation: send to peer review, not desk reject. The referee should push for a proof or precise attribution of the non-diagonal moderate deviations in Lemma 4, for the off-maximum estimates in Corollary 2, and for a prominent statement that the sharp coefficient is conditional on an open estimate.","headline":"A solid, honestly-flagged conditional result: the sharp √n coefficient rests on an unproved moderate-deviation range, but the unconditional theorems are real and worth serious refereeing.","tokens_in":30342,"tokens_out":2385,"would_cite":true,"duration_ms":28318,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F10","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"The most likely geodesic in last passage percolation is asymptotically the corner path that runs along the edges of the cube.","keywords":["last passage percolation","geodesics","corner path","moderate deviations","exponential weights","Bernoulli weights","monotonicity ordering","large deviations"],"falsifier":"Compute the point-to-line lower-tail constant in exponential LPP in the regime $t \\asymp n^{1/6}$: if $\\log P(L(0,0;n)\\le 2\\mu n - t n^{1/3})/t^3$ does not tend to $-1/48$, Assumption 3 fails and the matching limits in Theorem 2 are not the true asymptotics. Alternatively, for Bernoulli weights with $p\\le p_n$ (the explicit polynomial threshold in the proof of Theorem 3), enumerate all paths for small $n$ and check that every bump or unwind move strictly increases the geodesic probability; a counterexample refutes the monotonicity claim.","tokens_in":29345,"feed_emoji":"📐","tokens_out":32338,"duration_ms":245481,"temperature":0.7,"pith_summary":"The paper asks which path is most likely to be the geodesic in last passage percolation, and argues that the answer is an extreme corner path that runs straight along the edges of the cube. Under a sharp but not-yet-proved moderate deviation bound, it shows in two-dimensional exponential last passage percolation that the corner path's log-probability is $-2nI(\\mu)+\\frac{4}{3}\\sqrt{2n}+o(\\sqrt{n})$, and that this equals the maximum over all paths up to the $\\sqrt{n}$ order, making the corner path nearly modal. Unconditionally, it proves the corner path beats every path with linearly many corners by an exponential factor in any subcritical tiltable distribution, and for small Bernoulli weights it proves a monotonicity ordering under which the corner path is uniquely most likely. A sympathetic reader cares because these results turn a heuristic about entropic competition into explicit asymptotic formulae for the probability that a given path is the geodesic.","feed_headline":"The likeliest geodesic runs along the cube's corners","feed_subtitle":"The corner path matches the most likely geodesic through the √n term, with log probability about -2n(1-log2)+(4/3)√(2n).","key_machinery":"The central object is the corner path $\\gamma^\\ulcorner$, which for the cube $[0,n]^d$ is the path with the minimal number of corners ($d-1$). The quantitative results are carried by an exponent-balancing argument: the probability that a fixed path is the geodesic is bounded above by a sum over thresholds $a$ of $\\mathbb{P}(L(\\gamma)\\ge a)\\mathbb{P}(L_n\\le a+1)$, by negative correlation of the two monotone events. The first factor is the large-deviation tail of the path's own weight, $e^{-2nI(\\mu)+\\frac12 t n^{1/3}}$, and the second is the passage-time lower tail, $e^{-\\frac{1}{192}t^3}$, from the moderate deviation estimate (3). Maximizing $\\frac12 t n^{1/3} - \\frac{1}{192}t^3$ gives $t_* = 4\\sqrt{2}\\,n^{1/6}$ and the value $\\frac43\\sqrt{2n}$ that appears in the theorem. For the matching lower bound, the proof constructs an event in which every bulk shortcut is suppressed: the corner weights beat all point-to-line passage times (Lemma 3), and the point-to-line lower-tail bound is upgraded to a maximum over nearby starting points (Lemma 4), allowing the weights in the two endpoint segments of length $\\asymp n^{2/3}$ to be relaxed by an $O(n^{1/3})$ amount. The sharp point-to-line bound of Assumption 3 is exactly what makes this relaxation cost match the upper bound; without it, only the weaker constant $1/(3\\sqrt{3c_l})$ is obtained.","core_discovery":"In two-dimensional exponential last passage percolation, the paper's central claim is that the corner path $\\gamma^\\ulcorner$, which goes $(0,0)\\to(0,n)\\to(n,n)$, has probability of being the unique geodesic equal to $$\\log P(\\gamma^\\ulcorner = \\gamma^*_n) = -2n I(\\mu) + \\frac{4}{3}\\sqrt{2n} + o(\\sqrt{n}),$$ and that the same expression holds for $\\max_{\\gamma\\in\\Gamma_n}\\log P(\\gamma = \\gamma^*_n)$, so the corner path attains the maximum probability up to $o(\\sqrt{n})$ corrections. This is established conditionally on Assumption 3, a uniform sharp lower-tail moderate deviation bound for point-to-line passage times that is not currently proved for the needed range; the unconditional results give a limsup of $4/3$ for the normalized maximum and a liminf of $1/(3\\sqrt{3c_l})$ for the corner path, where $c_l>0$ is the point-to-line lower-tail constant. The paper also claims that for general tiltable subcritical weights in $\\mathbb{Z}^d$, any path with at least $\\epsilon n$ corners has probability at most $C_\\epsilon e^{-n/C_\\epsilon}$ times that of the corner path, and that for Bernoulli weights with sufficiently small $p$ in $d=2$, the two corner paths are exactly the unique most likely geodesics, with a monotonicity ordering (bump and unwind moves) that makes probabilities increase as a path is smoothed toward the corners.","pith_inferences":["The balancing exponent $t_* = 4\\sqrt{2}\\,n^{1/6}$ should be universal for the $\\sqrt{n}$ correction in exactly solvable LPP models; replacing the rate function $I$ and the moderate deviation constant $1/192$ by their geometric or Brownian counterparts yields the corresponding modal constant.","Lemma 4's method of upgrading a pointwise lower-tail bound to a maximum over a small number of correlated passage times at a super-exponential penalty is a transferable technique for problems in random growth models where many dependent passage times must be controlled simultaneously.","The monotonicity ordering for Bernoulli weights is proved only for polynomially small $p$; the signed-area formula for $C^{1,1}$ suggests the higher configuration counts $C^{a,b}$ with $a+b \\ge 3$ are the only obstruction, so a natural testable conjecture is that the same ordering holds for every subcritical $p<1/2$."],"forward_implications":["In two-dimensional exponential LPP, the probability that the corner path is the geodesic is asymptotically $\\exp(-2n(1-\\log 2) + \\frac{4}{3}\\sqrt{2n} + o(\\sqrt{n}))$, and the most likely geodesic has the same asymptotic probability up to $o(\\sqrt{n})$; the two corner paths are equally likely and dominate every other path at this order.","The set of paths whose geodesic probability is at least that of the corner path has size at most $e^{\\epsilon n}$ under the subcritical tiltability assumptions, whereas the total number of directed paths grows like $d^{dn}$; hence almost all paths are exponentially less likely than the corner path.","For Bernoulli weights with $p \\le p_n$, the corner path and its reflection are the unique most likely geodesics, and the partial order generated by bump and unwind moves is monotone: applying these moves can only increase the probability of being the geodesic.","Unconditionally, the normalized maximum log-probability satisfies $\\limsup_{n} (\\max_\\gamma \\log P(\\gamma=\\gamma^*_n)+2nI(\\mu))/\\sqrt{2n} \\le 4/3$, so the constant $4/3$ is an upper barrier that any sharper modal result must meet."],"supporting_citations":[{"why":"[Alb+25] supplies the lower bound $\\mathbb{P}(\\gamma^\\ulcorner \\in \\Gamma^*_n) \\ge \\exp(-dn(I(\\mu)+\\delta))$ in dimension 2, which Appendix B extends to all dimensions; this bound serves as the denominator in Theorem 1.","marker":"[Alb+25]"},{"why":"[Bas+25] gives sharp lower-tail moderate deviation estimates for beta ensembles; if extended to $t=O(n^{1/6})$ it would imply Assumption 3, and its half-space form provides the constant $c_h$ used in Corollary 2.","marker":"[Bas+25]"},{"why":"[Bas+21] establishes the point-to-line lower-tail moderate deviation bound (4) with an explicit positive constant $c_l$, which feeds the unconditional lower bound in Theorem 2 and Lemma 4.","marker":"[Bas+21]"},{"why":"[Bai+01] provides the point-to-point lower-tail moderate deviation expansion (3) for geometric weights, transferable to exponential weights; the paper uses it in the upper bound and in Lemma 4.","marker":"[Bai+01]"},{"why":"[BR60] supplies the tail refinement for sums of exponential random variables used in equation (2), which yields the factor $\\exp(-2nI(\\mu) + \\frac12 t n^{1/3})$ in the exponent balance.","marker":"[BR60]"},{"why":"[CZ03] gives the large-deviation lower-tail bound for the total passage time used as a negligible term in the proof of Theorem 1.","marker":"[CZ03]"},{"why":"[LR10] provides the small-deviation estimates behind the upper-tail passage-time bounds in Proposition 3 and the half-space tail constant used in Corollary 2.","marker":"[LR10]"},{"why":"[Zha20] gives the coalescence estimate used in Lemma 3 to split nearby passage-time maxima into independent blocks.","marker":"[Zha20]"}],"fun_headline_variants":["Corner path is the modal geodesic in last passage percolation","Straight to the corner: most likely geodesic in exponential LPP","Corner routes dominate geodesic probabilities in LPP","Most likely percolation geodesic hugs the cube's edge","In LPP, corner path attains max geodesic probability up to sqrt(n)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 3: the point-to-line lower-tail moderate deviation bound $\\log P(L(0,0;n)\\le 2\\mu n - t n^{1/3}) \\ge -\\frac{1}{48}t^3 + o(t^3)$ must hold uniformly for $T\\le t\\le \\rho n^{1/6+\\zeta}$, a range not covered by the best available estimates (which reach only $t=O(n^{1/10})$); if this bound is false or has a different constant, the matching limits in Theorem 2 and expression (5) collapse.","fun_headline_variants_meta":{"raw":{"variants":["Corner path is the modal geodesic in last passage percolation","Straight to the corner: most likely geodesic in exponential LPP","Corner routes dominate geodesic probabilities in LPP","Most likely percolation geodesic hugs the cube's edge","In LPP, corner path attains max geodesic probability up to sqrt(n)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1535,"prompt_tokens":999,"completion_tokens":536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":444}},"tokens_in":615,"tokens_out":536,"duration_ms":5054,"temperature":1.0,"reasoning_tokens":444,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:12:52.249305+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the point-to-line lower-tail constant in exponential LPP in the regime $t \\asymp n^{1/6}$: if $\\log P(L(0,0;n)\\le 2\\mu n - t n^{1/3})/t^3$ does not tend to $-1/48$, Assumption 3 fails and the matching limits in Theorem 2 are not the true asymptotics. Alternatively, for Bernoulli weights with $p\\le p_n$ (the explicit polynomial threshold in the proof of Theorem 3), enumerate all paths for small $n$ and check that every bump or unwind move strictly increases the geodesic probability; a counterexample refutes the monotonicity claim.","supporting_citations":[],"review_version":1}