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Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.09319 v2 pith:QOLKTH33 submitted 2019-08-25 math.PR

Elnur Emrah , Christopher Janjigian , Timo Seppäläinen This is my paper · ORCID
classification math.PR MSC 60K3560K37
keywords cornergrowthmodellast-passagepercolationlimitshapeinhomogeneousexponentialratesTASEPwithdisordervariationalformulaflatsegmentsspikesandcrevices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 4, Lemma 4.4] 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.
  2. [Section 5, proof of Theorem 3.2] 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).
minor comments (4)
  1. [Abstract and Section 1.1] The abstract contains a typo: 'addivitely' should be 'additively'.
  2. [Section 1.5 and Section 3.9] 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.
  3. [Section 3.6] 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.
  4. [Section 4, Lemma 4.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central variational centering is an explicit function of the model rates and is proved by self-contained concentration arguments.

full rationale

The paper's main centering claim, Theorem 3.2, is that G^{a,b}(m,n) is almost surely within smaller order than m+n of M^{a,b}(m,n), where M^{a,b}(m,n) is defined in (3.3) as the explicit infimum over z of sums of reciprocal rates. This quantity is not fitted to the data or to the target limit; it is a deterministic function of the parameters, and the proof proceeds through right-tail bounds (Lemma 4.2), geodesic exit probabilities (Lemma 4.6), and left-tail bounds (Lemmas 4.7 and 4.9). The shape function in Theorem 3.6 is then obtained by a deterministic approximation of the centering under vague convergence of empirical rate measures (Lemma 7.2), not by assuming the desired limit. The only non-elementary probabilistic input, the permutation invariance in Lemma 4.4, is quoted from the external paper [12], not from the authors' own prior work, and it is used to reduce the left-tail estimate to the nondecreasing parameter case. The Burke-type stationarity of increments, stated in Section 2.3, is cited to [6] for constant parameters and to [22] for the extension; [22] is a published, independent prior result of one of the authors, and the stationarity property itself is a testable structural fact rather than the paper's conclusion. Several auxiliary claims are deferred to the longer version [25], such as an omitted complementary upper bound in Remark 3.9.1 and some corollaries in Section 3.9, but these deferrals concern supporting remarks and secondary examples, not the main derivation chain of the centering or limit shape. No equation is defined in terms of the quantity it is supposed to predict, and no fitted parameter is renamed as a prediction. The paper is therefore self-contained against external benchmarks for its central claims, and the residual concerns about cited or omitted proofs are matters of verification and completeness, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results depend on the model definition, the imported permutation invariance lemma, and the convergence hypotheses in Theorem 3.6. There are no fitted free parameters or invented entities; the rate sequences a_i and b_j are arbitrary model inputs, not constants fitted to the target result.

assumptions (4)
  • domain assumption Waiting times are independent exponentials with rates a_i + b_j > 0 for all i,j (Section 1.4, Eq. (1.8)-(1.9)).
    This is the model under study; positivity ensures well-defined rates and makes the minimizer in (3.3) finite.
  • standard math Permutation invariance of G^{a,b}(m,n) under row and column permutations (Lemma 4.4), cited from the determinantal formula in [12, Eq. (12)].
    This imported result is used in Lemmas 4.6, 4.7, and 4.9 to remove a monotonicity assumption in the left-tail concentration estimates.
  • domain assumption Vague convergence of empirical rate distributions and convergence of running minima, as in (3.16)-(3.17), for the shape function and limit shape theorems.
    These are explicit hypotheses of Theorems 3.6 and 3.9; without them only the more general centering in Theorem 3.2 applies.
  • standard math Standard tools from probability and analysis, including concentration inequalities for sums of independent exponentials (Lemma A.2), vague convergence criteria (Lemma A.4), and Cauchy transform estimates (Lemma A.5).
    These are standard background results invoked in the proofs and stated in the appendix.

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Cite this review

Pith. "Pith review of Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model." pith.science (2026). https://pith.science/paper/QOLKTH33

@misc{pith2026190809319,
  author       = {Pith},
  title        = {Pith review of: Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QOLKTH33}},
  note         = {Machine review of arXiv:1908.09319}
}
read the original abstract

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Formulas of similar flavor were conjectured in arXiv:math/0004082, which we also establish. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

Figures

Figures reproduced from arXiv: 1908.09319 by the authors.

Figure 1
Figure 1. depicts a realization of R [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1.1
Figure 1.1. The cluster Rptq (red) and the boundary of the region tR (blue) at time t “ 1000 in four simulations of CGM with indicated rates. (a) Homo￾geneous case. R is given by (1.5) with c “ 1. (b) Flat spot above the diagonal (dashed gray) to the right of column 100 (dashed green) and spikes to the left of column 100. R is given by (1.6) in this and subsequent cases. (c) Larger spikes to the left of column 50 (dashed purple… view at source ↗
Figure 1.2
Figure 1.2. An illustration of the cluster along column m P Zą0 and after columns with much larger indices at time t (red) in the case min am ‰ a. The lines y ş R pb ` aq ´1βpdbq “ t (blue) and y ş R pb ` min amq ´1βpdbq “ t (purple) are shown. (a) A spike forms when min am ą a (b) A crevice forms when min am ă a. either omitted or postponed to Subsections 3.5-3.6. The statements pertinent to the vertical axis have obvious anal… view at source ↗
Figures from the paper (1 more)
Figure 1.3
Figure 1.3. Figure 1.3: An illustration of the boundary of the limit shape (blue) and the boundary of the region (1.18) (dashed gray). The strictly concave part and the (possibly empty) flat segments and spikes of the limit shape are indicated. flat segments can be understood geometrically …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.