{"id":"1b174463-498c-40a7-a0e1-14f197a78036","arxiv_id":"1908.09319","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the inhomogeneous exponential corner growth model, the paper proves explicit variational formulas for the growth asymptotics, identifies the limit shape with flat segments and spikes, and derives the disordered TASEP flux and height functions.","lead":"Some growing crystal-like blobs have random site-dependent speeds, and this paper finds exact formulas for their large-scale shape, including flat edges and spikes. It also settles a 2000 conjecture and gives the first full description of how such defects appear and persist in the corner growth model.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main results hinge on Lemma 4.4's permutation invariance, whose proof is only cited from [12]; if that cited formula does not cover the full parameter generality, the left-tail bound and all centering results lose their non-monotone case.","rationale":"The reader identified Lemma 4.4 as the weakest assumption, and I agree that this is the single most load-bearing external dependency. However, I do not find an actual error: the paper's concentration argument is internally consistent, the cited [12, Eq. (12)] is a published determinantal formula that is naturally symmetric in the rate parameters, and the use of Lemma 4.4 is limited to reducing to the monotone case in the left-tail estimates. The theorems are precisely stated, the Borel-Cantelli arguments are standard, and the limit-shape and TASEP consequences follow from the stated centering results. The concern is therefore about missing verification of a cited lemma, not about a demonstrated flaw. A direct check of the permutation symmetry for small m,n would settle the question without changing the accept verdict. I would keep the reader's ACCEPT verdict and moderate confidence, possibly encouraging the authors to include a short derivation or a more explicit citation in a revision.","tokens_in":43723,"tokens_out":22821,"duration_ms":231240,"concrete_test":"Independently verify Lemma 4.4 for small nontrivial sizes, e.g., m=n=3 with generic non-monotone admissible parameters a_i+b_j>0 satisfying (1.8). Compute the exact left tail P(G(3,3) <= x) using the determinantal formula [12, Eq. (12)] (or, as a numerical cross-check, a high-precision simulation with a fixed large sample of exponential weights), and compare it after applying an arbitrary row permutation and an arbitrary column permutation. If the distributions differ, the monotonicity reduction in Lemmas 4.7 and 4.9 is invalid and the first-order centering for arbitrary rates is not established. If they agree, the cited result supports the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.4 is the point where the paper imports its only integrable-probability input, and it is load-bearing. In the proof of Lemma 4.7, the text says 'By virtue of Lemma 4.4, the sequences (a_i) and (b_j) can be assumed to be nondecreasing without loss of generality,' and Lemma 4.9 repeats this reduction. Without monotonicity, the exit-probability bounds in Lemma 4.6 are unavailable, so the left-tail bound for G(m,n) below its centering is only proved in the monotone case. Theorem 3.2's lower bound, Corollary 3.3, Theorem 3.6, Theorem 3.9, and the TASEP corollaries all inherit this step. The proof of Lemma 4.4 is a single sentence citing [12, Eq. (12)]; it does not verify that the cited formula applies to real parameter arrays satisfying only (1.8), nor that it yields invariance under independent row and column permutations in the generality used here. This is a missing support rather than an observed internal contradiction: the rest of the concentration argument is coherent, and the cited determinantal formula is plausibly symmetric in the parameters. But because the entire non-monotone regime depends on this one unproved assertion, it is the most load-bearing assumption in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the exactly solvable inhomogeneous corner growth model with exponential waiting times whose rates at site (i,j) are a_m(i)+b_n(j), with parameter collections varying in the rectangle size. The central result, Theorem 3.2, gives an almost sure first-order centering: G^{a,b}(m,n) equals the explicit variational quantity M^{a,b}(m,n) = inf_z [sum_i 1/(a_m(i)+z) + sum_j 1/(b_n(j)-z)] up to errors of order o(m+n). Under vague convergence of the empirical parameter distributions and convergence of running minima, Theorem 3.6 identifies the shape function as gamma_{alpha,beta,a,b}(x,y) = inf_z [x A_alpha(z) + y B_beta(z)]. Theorem 3.9 describes the limit shape, including flat segments, spikes, and the persistence of macroscopic spikes and crevices that are invisible in the limit shape. The paper also resolves a conjecture of Rains in this setting (Theorem 3.4) and derives limiting height, flux, and particle position formulas for disordered TASEP (Theorems 3.10 and 3.11). The method is based on concentration bounds, stationary-increment couplings, and boundary exit estimates for geodesics; the only integrable-probability input is the distributional invariance of the last-passage times under permutation of the parameters, stated as Lemma 4.4.","tokens_in":44011,"tokens_out":4142,"duration_ms":45644,"significance":"If the main theorems are fully established, this is a substantial contribution: it provides explicit, parameter-free variational formulas for the a.s. leading-order behavior of a non-stationary exactly solvable growth model, confirms a conjecture from Rains, and gives a detailed and partly surprising description of the limit shape, including flat segments and spikes alongside macroscopic features not visible in the limit. The concentration estimates and exit-probability arguments are worked out in detail, and the paper is careful to state the mild growth conditions under which the centering claims hold. The TASEP applications give useful explicit height and flux centerings in the presence of particlewise and holewise disorder. The main caveat is that one imported statement, Lemma 4.4, is load-bearing for all non-monotone parameter results and is not proved in the manuscript; this makes the current version conditional on the correctness and applicability of the cited determinantal formula.","major_comments":[{"comment":"Lemma 4.4 is the only integrable-probability input in the paper and is used in a central way: in the proof of Lemma 4.7 the text states that 'by virtue of Lemma 4.4, the sequences (a_i) and (b_j) can be assumed to be nondecreasing without loss of generality', and Lemma 4.9 repeats this reduction. This monotone reduction is needed before the exit-probability bounds of Lemma 4.6 can be applied, and it ultimately supports the lower bound in Theorem 3.2, Corollary 3.3, Theorem 3.6, and Theorem 3.9. However, the proof of Lemma 4.4 is a single sentence citing equation (12) of [12], without verifying that the cited determinantal formula indeed applies to the two-index parameter collections in (1.7)-(1.8) and that it yields invariance under independent row and column permutations in the full generality needed here. This is a missing justification rather than an observed contradiction, but because the entire non-monotone regime depends on this assertion, the authors should either provide a self-contained proof of Lemma 4.4 or state the precise theorem from [12] with all hypotheses checked, including the regularity conditions on the parameters.","section":"Section 4, Lemma 4.4"},{"comment":"The lower-bound part of Theorem 3.2 invokes Lemma 4.9, whose proof explicitly depends on Lemma 4.4 in the same way as Lemma 4.7 does. Thus the non-monotone case of the paper's central centering result is not independently established within the manuscript; it inherits the unresolved status of Lemma 4.4. If Lemma 4.4 is correct and applicable, the argument appears coherent, but as written the proof of the main theorem is incomplete for general parameter arrays satisfying only (1.8).","section":"Section 5, proof of Theorem 3.2"}],"minor_comments":[{"comment":"The abstract contains a typo: 'addivitely' should be 'additively'.","section":"Abstract and Section 1.1"},{"comment":"Several corollaries and computations, including some parts of the flat-segment description and connections to earlier models, are deferred to the longer version [25]. The journal version should ensure that all claims stated as results are either proved in the text or accompanied by precise references to the longer version.","section":"Section 1.5 and Section 3.9"},{"comment":"The parametrization of the curved part of the limit shape is stated without a displayed derivation; adding the short computation that Phi(z) parametrizes the boundary would improve readability.","section":"Section 3.6"},{"comment":"In the proof of Lemma 4.5, after the choice z = zeta - c(k-1)Delta/m, the constraint c < 1/3 is only introduced at the end; stating it before the estimate would make the argument easier to follow.","section":"Section 4, Lemma 4.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically rich and the main results are plausible, but Lemma 4.4 is a single load-bearing import from [12] and is invoked to remove the monotonicity assumption in the central concentration argument. I would ask the authors to prove Lemma 4.4 or to give a precise statement of the external result with all hypotheses verified before publication. I saw no evidence of circular reasoning or fitted parameters; the centering is an explicit infimum of reciprocals of the model rates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper earns its length. The main theorems are exactly what they claim to be: explicit variational centering for the inhomogeneous exponential corner growth model, resolution of Rains' conjecture, and a limit-shape description with flat segments, spikes, and crevices. The nonsubadditive route via stationary increments and concentration bounds is well executed, and the new macroscopic phenomena (persistent spikes and crevices not visible in the limit shape) are interesting and convincingly demonstrated. The centering (3.3) is genuinely parameter-free, and the transfer to TASEP is a useful bonus.\n\nThe proofs are careful and I found no internal contradiction in the concentration argument. The comparison in Lemma 7.2 and the geodesic exit bounds in Lemmas 4.5–4.6 are nontrivial and look correct. Lemma 4.1 is a routine exponential concentration bound; the harder left-tail work in Lemmas 4.7 and 4.9 checks out, conditional on its stated inputs.\n\nThe one real soft spot is Lemma 4.4, which asserts distributional invariance of last-passage times under independent permutations of the row and column parameters. This is cited from [12, Eq. (12)] without proof, and it is load-bearing: it is how the left-tail bound is extended from nondecreasing to arbitrary parameters, and Theorem 3.2, Corollary 3.3, Theorem 3.6, Theorem 3.9, and all TASEP corollaries inherit this step. The stress-test note worries that the cited formula may not cover real parameter arrays satisfying only (1.8). I think that concern is legitimate as a request for support, not as evidence of a mistake—the Borodin–Peché formula is symmetric in the parameters and the claim is plausibly true in this generality. But because the entire non-monotone regime rests on this one sentence, a referee should insist on either a proof or a precise statement of the cited result that verifies the hypotheses.\n\nMinor quibbles: the vague-convergence assumptions in Theorem 3.6 are natural but could be stated as sufficient rather than necessary, and the simulations in Section 1 are illustrative rather than rigorous evidence. Neither affects the mathematical claims.\n\nIn short: this is a strong paper that deserves a serious referee. I would send it out, with the instruction that the referee verify or repair Lemma 4.4. If that lemma holds up, the paper is ready for a top probability journal.","headline":"A rigorous, explicit resolution of the Rains conjecture for inhomogeneous exponential LPP with a genuinely new description of spikes and crevices; the only real soft spot is a load-bearing but unproved permutation-invariance lemma imported from a cited paper.","tokens_in":44521,"tokens_out":1528,"would_cite":true,"duration_ms":18170,"reading_group":"yes","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":"The paper proves explicit variational first-order asymptotics for the inhomogeneous exponential corner growth model and describes the limit shape with flats, spikes, and hidden crevices.","keywords":["corner growth model","last-passage percolation","limit shape","inhomogeneous exponential rates","TASEP with disorder","variational formula","flat segments","spikes and crevices"],"falsifier":"For $m=n=2$, compute the exact distribution of $G^{a,b}(2,2)$ from the explicit formula cited in the paper for two choices of parameters, e.g. $(a_1,a_2)=(1,2)$, $(b_1,b_2)=(3,4)$, and for the permuted choices $(a_1,a_2)=(2,1)$, $(b_1,b_2)=(4,3)$. If the two distributions differ, the permutation-invariance lemma used to remove monotonicity in the left-tail bound is false, and the proof of Theorem 3.2 does not cover arbitrary rate parameters.","tokens_in":43549,"feed_emoji":"📈","tokens_out":8949,"duration_ms":89461,"temperature":0.7,"pith_summary":"The paper establishes deterministic first-order formulas for the inhomogeneous corner growth model, the last-passage percolation process built from independent exponential waiting times whose rates are sums of a column parameter and a row parameter. Its central theorem says that once the minimal total rate does not decay too fast, the growth time at site $(m,n)$ is almost surely equal, to leading order in $m+n$, to the explicit infimum $\\inf_{z\\in(-\\min a_m, \\min b_n)}\\big(\\sum_i (a_m(i)+z)^{-1} + \\sum_j (b_n(j)-z)^{-1}\\big)$. Under natural limits of the empirical rate distributions, this infimum becomes a shape function, so the limiting cluster is an explicit set whose boundary can develop flat segments, axis spikes, and even persistent spikes and crevices invisible in the limit. The formulas also give the flux function and particle profile for TASEP with step initial data and site disorder. This matters because local inhomogeneities are shown to reshape the limit shape and the macroscopic dynamics in a fully computable way.","feed_headline":"Explicit formula gives flats, spikes and crevices in growth","feed_subtitle":"Variational shape function pins down disorder effects in corner growth and in TASEP with site disorder.","key_machinery":"The argument is carried by the increment-stationary last-passage processes $\\tilde G^{a,b,z}$: couplings in which, for each $z$ in $(-\\min a_m, \\min b_n)$, the horizontal increments in a row are independent exponentials with rates $a_m(i)+z$ and the vertical increments in a column are independent exponentials with rates $b_n(j)-z$, so increments are stationary under translations (the Burke property). The mean of $\\tilde G^{a,b,z}(m,n)$ is $M^{a,b,z}(m,n)$, and the true process is sandwiched around the unique minimizer $z$ of this convex function. Summable concentration bounds are obtained from exponential tail estimates for sums of independent exponentials and from controlling the probability that geodesics exit the rectangle through its boundary far from the origin. The only input from determinantal structure is the lemma, read off a cited distributional formula, that the law of $G^{a,b}(m,n)$ is invariant under permutations of the $a_m(i)$'s and $b_n(j)$'s; this invariance is what removes monotonicity assumptions from the left-tail bound.","core_discovery":"Let $G^{a,b}(m,n)$ be the largest total weight of an up-right path from $(1,1)$ to $(m,n)$ when the independent weights are exponential with rate $a_m(i)+b_n(j)$. The paper proves that, $P$-a.s., for all large $m+n$, $G^{a,b}(m,n)$ equals $M^{a,b}(m,n)=\\inf_{z\\in(-\\min a_m, \\min b_n)}\\big(\\sum_{i=1}^m (a_m(i)+z)^{-1} + \\sum_{j=1}^n (b_n(j)-z)^{-1}\\big)$ up to an error of smaller order than $m+n$, with explicit upper and lower tail bounds. When the empirical distributions of $a_m$ and $b_n$ converge vaguely to subprobability measures $\\alpha$ and $\\beta$ and the running minima converge to $a$ and $b$, the same result holds with the shape function $\\gamma_{\\alpha,\\beta,a,b}(x,y)=\\inf_{z\\in(-a,b)}\\big(x\\int \\alpha(da)/(a+z)+ y\\int \\beta(db)/(b-z)\\big)$. The limit shape is then the sublevel set of this function together with explicit axis segments, and the paper describes exactly when its boundary is strictly concave, flat, or spiked. These results also prove the older variational conjecture formulated as Theorem 3.4 and imply explicit centerings for the disordered TASEP height and flux.","pith_inferences":["The same coupling-plus-variational strategy would likely transfer to other exactly solvable last-passage models that possess an analogous permutation-invariance property, even when full kernel asymptotics are unavailable.","Because flat-segment formation is governed by the tail of $\\alpha$ near its infimum, one can test for a geometric phase transition by tuning the decay of column parameters; the paper does not compute fluctuation orders across this transition.","The fact that crevices vanish from the limit shape suggests that first-order shape functions miss a whole macroscopic structure; natural follow-up statistics are columnwise overshoot and undershoot magnitudes, and entropy of the height profile.","Finite-size simulations against $t\\mathcal{R}$ should reveal the predicted spike and crevice intervals; the paper gives such illustrations but not a quantitative goodness-of-fit criterion."],"forward_implications":["For any inhomogeneous exponential corner growth model satisfying the mild growth condition, leading-order growth values are computed by a one-dimensional convex minimization; the result does not require subadditive ergodic theory.","When empirical rate distributions and running minima converge, the limit shape is completely explicit: the sublevel set of the shape function plus axis segments, with strictly concave boundary in one cone and flat boundary in the complementary regions.","Axis spikes in the limit shape occur exactly when the supremum of the running minima exceeds its limit value, and flat segments adjacent to an axis occur exactly when a certain reciprocal-square integral is finite.","Macroscopic spikes and crevices form along columns whose minimum rate is respectively above or below the limiting minimum; the limit shape encodes only the maximal spikes, not the crevices or smaller spikes.","For TASEP with step initial condition and particle- and holewise disorder, the flux function and limiting particle profile are given by explicit formulas, so disorder effects on the current are computable."],"supporting_citations":[{"why":"Supplies the determinantal formula whose consequence, permutation invariance of the distribution of G, is the only integrable-probability input used to remove monotonicity in the left-tail bound.","marker":"[12]"},{"why":"The preprint whose conjectured variational limit is reformulated and proved as Theorem 3.4.","marker":"[46]"},{"why":"Earlier derivation of the same shape-function approximation under ergodicity and along fixed directions; the present theorem strengthens it to full a.s. first-order asymptotics.","marker":"[22]"},{"why":"Establishes the homogeneous parabolic limit shape that serves as the baseline for the inhomogeneous results.","marker":"[48]"},{"why":"Gives the increment-stationary (Burke) version for constant parameters, which the present paper extends to inhomogeneous additive rates.","marker":"[6]"},{"why":"Introduces the exactly solvable exponential corner growth model with growing rates, one of the settings unified here.","marker":"[36]"}],"fun_headline_variants":["Explicit formula shows flats, spikes, crevices in growth","Corner growth shape: flats, spikes, crevices from explicit formula","Exact shape for disordered corner growth with spikes and flats","Variational formula exposes flats, spikes, crevices in growth","Flats, spikes, crevices: explicit shape for corner growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the left-tail half of the main centering theorem relies on the unproved claim, quoted from a determinantal formula in the cited literature, that the law of $G^{a,b}(m,n)$ is unchanged when the row or column rate parameters are permuted; if that invariance failed, the centering theorem would only be established for monotone parameter sequences.","fun_headline_variants_meta":{"raw":{"variants":["Explicit formula shows flats, spikes, crevices in growth","Corner growth shape: flats, spikes, crevices from explicit formula","Exact shape for disordered corner growth with spikes and flats","Variational formula exposes flats, spikes, crevices in growth","Flats, spikes, crevices: explicit shape for corner growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1475,"prompt_tokens":1138,"completion_tokens":337,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":754,"completion_tokens_details":{"reasoning_tokens":249}},"tokens_in":754,"tokens_out":337,"duration_ms":3706,"temperature":1.0,"reasoning_tokens":249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:14:49.728543+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $m=n=2$, compute the exact distribution of $G^{a,b}(2,2)$ from the explicit formula cited in the paper for two choices of parameters, e.g. $(a_1,a_2)=(1,2)$, $(b_1,b_2)=(3,4)$, and for the permuted choices $(a_1,a_2)=(2,1)$, $(b_1,b_2)=(4,3)$. If the two distributions differ, the permutation-invariance lemma used to remove monotonicity in the left-tail bound is false, and the proof of Theorem 3.2 does not cover arbitrary rate parameters.","supporting_citations":[{"cited_title":"Borodin and S","cited_arxiv_id":null,"evidence_quote":"Supplies the determinantal formula whose consequence, permutation invariance of the distribution of G, is the only integrable-probability input used to remove monotonicity in the left-tail bound."},{"cited_title":"A mean identity for longest increasing subsequence problems","cited_arxiv_id":"math/0004082","evidence_quote":"The preprint whose conjectured variational limit is reformulated and proved as Theorem 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier derivation of the same shape-function approximation under ergodicity and along fixed directions; the present theorem strengthens it to full a.s. first-order asymptotics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the homogeneous parabolic limit shape that serves as the baseline for the inhomogeneous results."},{"cited_title":"Bal´ azs, E","cited_arxiv_id":null,"evidence_quote":"Gives the increment-stationary (Burke) version for constant parameters, which the present paper extends to inhomogeneous additive rates."},{"cited_title":"Johansson","cited_arxiv_id":null,"evidence_quote":"Introduces the exactly solvable exponential corner growth model with growing rates, one of the settings unified here."}],"review_version":1}