KP governs random growth off a one dimensional substrate

Daniel Remenik; Jeremy Quastel

REVIEW 2 major objections 4 minor 2 cited by

KP governs random growth off a one dimensional substrate

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The KP equation governs the marginals of random growth interfaces in one dimension.

desk verdict A genuinely new and well-supported connection between the KPZ fixed point and the KP equation; the derivation is sound and the main soft spot is the title's universality claim, which the body itself flags as conjectural. read the letter →

arxiv 1908.10353 v5 pith:WVRTNWM2 submitted 2019-08-27 math.PR math-phmath.MP

Jeremy Quastel , Daniel Remenik This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3535Q5337K10

keywords KPZuniversalityclassfixedpointKadomtsev-PetviashviliequationFredholmdeterminantsTracy-WidomdistributionsAiryprocessintegrabilityrandomgrowth

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 that the logarithmic derivative of the finite-dimensional marginals of the KPZ fixed point—the conjectural universal scaling limit of one-dimensional random growth—satisfies the Kadomtsev–Petviashvili (KP) equation. Concretely, for an $n$-point height distribution $F$, the matrix $Q$ built from the Fredholm determinant kernel obeys the matrix KP equation, and $D_r \log F$ equals $\operatorname{tr} Q$; for one-point marginals the equation reduces to scalar KP-II. This is shown by algebra from the kernel of the Fredholm determinant for the KPZ fixed point, without a physical heuristic. As corollaries, the Tracy–Widom GUE and GOE distributions appear as self-similar solutions of KP/KdV, and several known exact solutions of the KPZ equation also satisfy KP-II. The result gives the scaling limit a closed PDE structure that it was not known to have.

What carries the argument

The load-bearing object is the extended Brownian scattering operator for the KPZ fixed point, defined through Brownian motion killed when it hits the hypograph of the initial height, conjugated by the Airy unitary group $U_t=e^{-t\partial^3/3}$. Its kernel $K$ obeys three differential relations—$D_r K=(D_1+D_2)K$, $\partial_t K=-\tfrac13(D_1^3+D_2^3)K$, and $D_x K=(D_2^2-D_1^2)K$—together with an integration-by-parts identity $[A][B]=-[A D_1 B + D_2 A B]$ for matrix entries evaluated at $(0,0)$. These relations are what force $Q=[(I-K)^{-1}K]$ to satisfy the matrix KP equation; the scalar version emerges by taking traces. The same relations also characterize which Fredholm determinants are KP tau functions, connecting the stochastic growth problem to the KP hierarchy.

What would settle it

Compute the 1:2:3-scaled two-point distribution function $F$ of a growth model that is believed to lie in the KPZ universality class but is not known to converge to the same fixed point as TASEP, and check numerically or exactly whether $\partial_r^2 \log F$ satisfies the scalar KP-II equation (1.7); a nonzero residual would falsify the claim that KP governs random growth off a one-dimensional substrate.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: let $K$ be the shifted extended Brownian scattering kernel for the KPZ fixed point, $R=(I-K)^{-1}$, and $Q=[RK]$ the $n\times n$ matrix of $RK$ evaluated at $(0,0)$; then $Q$ and $q=D_r Q$ solve the matrix KP equation $\partial_t q + \tfrac12 D_r q^2 + \tfrac1{12} D_r^3 q + \tfrac14 D_x^2 Q + \tfrac12[q,D_x Q]=0$, and the logarithmic derivative of the $n$-point distribution is $D_r \log F = \operatorname{tr} Q$. In the one-point case, $\varphi=D_r^2 \log F$ solves the scalar KP-II equation (1.7). The authors emphasize that the result was unexpected and follows essentially by algebra from the kernel's differential relations; they also recover the GUE and GOE Tracy–Widom laws as self-similar solutions and show that several explicit KPZ-equation solutions (narrow wedge, spiked, and two-sided Brownian initial data) satisfy KP-II.

Load-bearing premise

The broad statement that KP governs random growth off a one-dimensional substrate relies on the unproven universality of the KPZ fixed point: the theorem is proved for the fixed point obtained as the 1:2:3 scaling limit of TASEP, and if some model in the class has a different large-scale limit, the general claim does not follow from this derivation.

Editorial extensions

If this is right

The GUE and GOE Tracy–Widom distributions appear as self-similar solutions of KP/KdV, so the KP equation supplies the 1:2:3 scaling invariance that the Tracy–Widom laws themselves lack.
The multipoint distribution of the Airy$_2$ process satisfies an explicit PDE (Theorem 1.9), giving a partial answer to the longstanding question of whether such a closed equation exists.
For flat initial data, the integrated matrix KdV equation governs the multipoint Airy$_1$ distribution, placing the flat case in the same integrable-systems framework.
Several known exact solutions of the KPZ equation, including narrow-wedge, spiked/half-Brownian, and two-sided Brownian initial data, are shown to produce KP-II solutions for their logarithmic derivatives.
The lower tails of the Tracy–Widom distributions can be recovered directly from the Burgers part of KP, which dominates in that regime.

Reading between the lines

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

Editorial inference: because the proof uses only the three kernel differential relations, any determinantal stochastic process whose kernel satisfies those relations will automatically produce KP solutions; this gives a testable algebraic criterion for discovering new integrable structures in random growth.
Editorial inference: the emergence of matrix KP suggests the full finite-dimensional transition probabilities of the KPZ fixed point may be tau functions of the KP hierarchy, in which case higher-order hierarchy equations would impose additional constraints on multipoint distributions beyond (1.6).
Editorial inference: if the broad universality claim holds, the KP equation becomes a practical numerical tool for approximating growth-model distributions at large scales—solving KP with escarpment initial data could yield predictions for tails and correlations that are currently obtained only from Fredholm determinant expansions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central theorem is real and the derivation holds up. What the paper proves is that the finite-dimensional marginals of the KPZ fixed point, starting from the MQR17 Fredholm determinant, satisfy matrix and scalar KP. The mechanism is clean: the fixed point kernel satisfies the differential relations (3.7)/(3.9), and once you have those, the Fredholm-determinant-to-KP machinery (Pöppe, going back to Zakharov–Shabat) does the rest. The new content is that the kernel actually evolves that way and that random growth is thereby connected to integrable systems; the Tracy-Widom distributions reappearing as self-similar solutions of KP is a satisfying payoff, not an input. There is no fitting anywhere. The proof is honest: the equation is checked for large r where trace norms are small, then extended by real analyticity, and the paper says which parts are imported from where. The acknowledgement of the v1 flat-initial-data error is a good sign; it also shows the mechanism is delicate, and the paper does not lean on it.

Soft spots, in proportion. The title and the abstract's opening sentence claim more than the theorem: what is proved is that the TASEP-derived fixed point satisfies KP. The step from there to "randomly fluctuating interfaces in one dimension" is the standard universality conjecture, which the paper explicitly labels as widely believed rather than proved. That is a presentation issue, not a mathematical gap — the body is careful. The second soft spot is the initial data: the escarpment data (1.9) is formal, and the authors state plainly that the matrix initial data (1.10) is insufficient as written because the 0 and ∞ entries interact. Well-posedness of the PDE is left open. That does not threaten Theorem 1.1, but it does mean the PDE side of the story is less complete than the probability side. Minor items: the abstract says "logarithmic derivative" where the scalar result is for the second r-derivative; the relation to the Adler–van Moerbeke/Tracy–Widom equations for the Airy process is left unresolved; and the Section 3.3 computation is condensed enough that a referee should actually check the cancellations, especially in the matrix case.

Who is this for: anyone working on KPZ universality or integrable probability. It deserves a serious referee. I would send it out, with a request that the authors soften the universality language and keep the honest caveats.

Referee Report

2 major / 4 minor

Summary. The paper proves that the logarithmic derivatives of the finite-dimensional distributions of the KPZ fixed point satisfy the matrix Kadomtsev-Petviashvili (KP) equation, and that in the one-point scalar case the second logarithmic derivative satisfies scalar KP-II. The proof starts from the Fredholm determinant formula for the KPZ fixed point obtained in [MQR17] as the 1:2:3 limit of TASEP, rewrites the kernel so that derivatives in t, x, and r act through the differential relations (3.7) and (3.9), computes the logarithmic derivative as tr Q, and then performs a lengthy but explicit operator-algebraic calculation to derive (1.6). Rigorous justification is by trace-class estimates for sufficiently large r followed by real analytic continuation. The paper also presents the Tracy-Widom distributions as self-similar solutions, formal initial data for escarpments, equations for Airy processes, and several KPZ-equation special solutions satisfying KP-II.

Significance. If the main theorem is correct, it establishes a new and surprising link between the integrable probability of the KPZ fixed point and integrable PDE theory. The derivation is honest and explicit: the relevant differential relations are stated, the analytic-continuation argument is described, and the paper flags precisely where rigorous support is imported from earlier work. The examples and the lower-tail heuristics are valuable. The main limitation is that the theorem is proved for the TASEP-derived KPZ fixed point, so the title and abstract's general statement about all random growth in one dimension rests on the unproven KPZ universality conjecture; the authors state this themselves. This does not undermine Theorem 1.1 but means the advertised scope should be qualified.

major comments (2)

[Title, Abstract, and Section 1 after (1.2)] The general claim that KP governs random growth off a one-dimensional substrate is wider than what is proved. Theorem 1.1 is established for the KPZ fixed point defined through the TASEP 1:2:3 limit in [MQR17], and the only support for the universal statement is the sentence after (1.2): 'It is widely believed that this KPZ fixed point governs the limiting fluctuation for all models in the class.' Since universality is not proved here, the title and abstract overstate the result. This is not an internal inconsistency, but it is load-bearing for the advertised message. Please qualify the statements, for example by saying 'for the TASEP-derived KPZ fixed point' or 'for models in the class where the same fixed point has been proved.'
[Section 1.1, Eqs. (1.9) and (1.10)] The paper's language that distributions 'evolve according to' KP is stronger than the theorem. Theorem 1.1 is a pointwise differential identity for Q and q at t > 0; it does not establish well-posedness of the initial-value problem. The text after (1.9) explicitly says that well-posedness with escarpment initial data is left for future work, and the matrix initial data (1.10) is stated to be insufficient because 'the 0 and ∞ interact' and would need augmentation by 'some description of the rate of convergence.' These caveats should be reflected in the abstract or in the statement of the main claim so that 'evolve according to KP' is not read as a fully justified evolution statement.

minor comments (4)

[Abstract and Section 1, Eq. (1.7)] The abstract's first sentence says the logarithmic derivative evolves by KP, but the scalar equation (1.7) is for φ = ∂_r^2 log F, not for ∂_r log F itself. The body is precise; please correct the abstract wording.
[Section 1.4, Eqs. (1.17) and (1.18)] The paper states that the new Airy-process equation and the Adler-van Moerbeke equation 'do not appear to be equivalent' and leaves the reconciliation for future work. This is acceptable, but a brief explanation of whether a discrepancy is suspected or merely a different form would help readers.
[Section 1.1, after Eq. (1.9)] The phrase 'the ∞ looks formal' is ambiguous; likely 'the −∞ in the initial data looks formal' is intended. Please clarify.
[References] The reference [MQR+] is listed as 'In preparation'; if an arXiv or published version exists, please update the citation so readers can access it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is a genuine derivation from the MQR17 Fredholm determinant, with no fitted parameter or assumed KP equation; the remaining universality gap is a scope concern, not a circular one.

full rationale

The paper's central derivation starts from the explicit Fredholm determinant representation (3.6) taken from [MQR17], a prior theorem by the same authors that establishes the KPZ fixed point formula from the TASEP 1:2:3 limit. That prior result does not contain the KP statement, so citing it is not circular: it is an independent input with stated assumptions that do not include the target equation. The proof then defines Q = [RK] in (3.8), derives the kernel differential relations (3.7) and (3.9) from the Brownian scattering transform and the Airy semigroup definitions, and performs a long operator calculation to obtain the matrix KP equation (1.6). The scalar KP-II equation (1.7) is a corollary, not an input. There are no fitted parameters, no normalization constants tuned to the target result, and no use of the conclusion as an assumption. The Tracy-Widom examples are also not circular: they insert a self-similar ansatz into the derived KP equation, reduce to the Painleve II ODE, and identify the Hastings-McLeod solution using standard asymptotics; the identification with established Tracy-Widom laws is an application, not a premise. Likewise, Section 2 explicitly verifies that known explicit kernels from the literature satisfy the same differential relations, and the authors honestly state that 'All we have is examples' and that they do not know whether the KP property extends generally. The only load-bearing gap between Theorem 1.1 and the abstract's sweeping statement about all one-dimensional random growth is the unproven universality of the KPZ fixed point, expressed by the sentence 'It is widely believed that this KPZ fixed point governs the limiting fluctuation for all models in the class.' That is a conjecture about scope, not a circular derivation: even if universality failed, Theorem 1.1 would remain valid for the TASEP-derived fixed point. The acknowledgement retracting an earlier KdV claim for flat initial data further shows that the authors treat the algebraic mechanism as delicate and do not conceal failures. Overall, no step in the paper reduces by definition, by fitted input, or by a self-citation chain to its own conclusion, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation uses no fitted parameters. It relies on the determinantal formula for the KPZ fixed point and on analyticity properties imported from earlier papers, plus the unproven universality assumption for the broad interpretation.

assumptions (5)

domain assumption The Fredholm determinant formula (3.6) from MQR17 correctly represents the finite-dimensional distributions of the KPZ fixed point for the class of initial data considered.
The entire derivation in Section 3 starts from this formula; the paper does not rederive it.
domain assumption The extended kernel K satisfies the differential relations ∂tK = -(1/3)(D1^3 + D2^3)K and DxK = (D2^2 - D1^2)K, stated as (3.9).
These relations are used throughout the algebraic computation in Section 3.3 and are asserted to follow from the definition of the Brownian scattering transform.
domain assumption The Fredholm determinant det(I-K) is real analytic and non-vanishing for t > 0 and finite x, r, so the KP equation proved for large r extends to all r by analytic continuation.
Remark 3.2 relies on trace class estimates from MFQR13 and the lower bound from MQR17, Prop. 4.7; the paper does not reproduce these proofs.
domain assumption The KPZ fixed point is the universal 1:2:3 scaling limit for all models in the one-dimensional KPZ universality class.
Stated as 'widely believed' in the introduction; the theorem is proved for the fixed point obtained from TASEP, not for every model in the class.
ad hoc to paper The KP equation with escarpment initial data (1.9) is well posed.
The authors write, in Section 1.1 after (1.9), 'we believe the solutions to the equations with such initial data are well posed, but we leave the proofs for future work.' This is needed if KP is to be interpreted as a governing evolution equation from the substrate.

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Cite this review

Pith. "Pith review of KP governs random growth off a one dimensional substrate." pith.science (2026). https://pith.science/paper/WVRTNWM2

@misc{pith2026190810353,
  author       = {Pith},
  title        = {Pith review of: KP governs random growth off a one dimensional substrate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVRTNWM2}},
  note         = {Machine review of arXiv:1908.10353}
}

read the original abstract

The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev-Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17, arXiv:1701.00018] for the KPZ fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy-Widom distributions appear as special self-similar solutions of KP and KdV. In addition, it is noted that several known exact solutions of the KPZ equation also solve KP.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Riemann surfaces for KPZ with periodic boundaries
cond-mat.stat-mech 2019-08 conditional novelty 6.0 of 10

Known exact finite-volume KPZ fluctuation probabilities are expressed as traces on Riemann surfaces for half-integer polylogarithms, and prior formulas by Prolhac and by Baik and Liu are proved equivalent.
Dynamic scaling of growing interfaces
cond-mat.dis-nn 2025-07 unverdicted novelty 1.0 of 10

An expert review of the KPZ equation's history, mathematical developments, and applications, with no new research results.

Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages · cited by 2 Pith papers

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Limiting one-point distribution of periodic TASEP

arXiv: 2008.07024 [math.PR]. [BCF14] A. Borodin, I. Corwin, and P. Ferrari. Free energy ﬂuctuations for directed polymers in random media in 1 + 1 dimension. Comm. Pure Appl. Math. 67.7 (2014), pp. 1129–1214. [BCFV15] A. Borodin, I. Corwin, P. Ferrari, and B. Vet ˝o. Height ﬂuctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18.1 (2015), Art. 20,

work page Pith review arXiv 2014

Pith tools

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

[1] [1]

[AS79] M. J. Ablowitz and H. Segur. On the evolution of packets of water waves. Journal of Fluid Mechanics 92.4 (1979), pp. 691–715. [AM05] M. Adler and P. van Moerbeke. PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab. 33.4 (2005), pp. 1326–1361. [ACQ11] G. Amir, I. Corwin, and J. Quastel. Probability distribution of th...

work page 1979

[2] [7]

Solution of the Kolmogorov equation for TASEP

arXiv: 1906.01692 [math.PR]. [NQR20] M. Nica, J. Quastel, and D. Remenik. One-sided reﬂected Brownian motions and the KPZ ﬁxed point

work page Pith review arXiv 1906

[3] [8]

[Po89] C

arXiv: 2002.02922 [math.PR]. [Po89] C. Pöppe. General determinants and theτ function for the Kadomtsev-Petviashvili hierar- chy. Inverse Problems 5.4 (1989), pp. 613–630. [Pro20] S. Prolhac. Riemann surfaces for KPZ with periodic boundaries. SciPost Phys. 8 (2020). [QR19] J. Quastel and D. Remenik. How ﬂat is ﬂat in random interface growth? Trans. Amer. M...

arXiv 1989

[4] [95]

Borodin, P

[BFPS07] A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial conﬁguration. J. Stat. Phys. 129.5-6 (2007), pp. 1055–1080. [CDR10] P. Calabrese, P. L. Doussal, and A. Rosso. Free-energy distribution of the directed polymer at high temperature. EPL (Europhysics Letters) 90.2 (2010), p. 20002. [CF...

work page 2007

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Differential equations, symmetries and inﬁnite-dimensional algebras, Translated from the 1993 Japanese original by Miles Reid

Cambridge Tracts in Mathematics. Differential equations, symmetries and inﬁnite-dimensional algebras, Translated from the 1993 Japanese original by Miles Reid. Cambridge University Press, Cambridge, 2000, pp. x+108. [MFQR13] G. Moreno Flores, J. Quastel, and D. Remenik. Endpoint distribution of directed polymers in 1 + 1 dimensions. Comm. Math. Phys. 317....

work page 2013

[6] [1812]

[Dot10] V

00309 [math.PR]. [Dot10] V . Dotsenko. Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. EPL (Europhysics Letters) 90.2 (2010), p. 20003. [IS12] T. Imamura and T. Sasamoto. Exact solution for the stationary Kardar-Parisi-Zhang equa- tion. Phys. Rev. Lett. 108 (19 2012), p. 190603. KP(Z) 20 [IS13] T. Imamura and...

work page 2010

[7] [2019]

Large deviations for the KPZ equation from the KP equation

arXiv: 1910.03671 [cond-mat]. [LDC12] P. Le Doussal and P. Calabrese. The KPZ equation with ﬂat initial condition and the directed polymer with one free end. J. Stat. Mech. 2012.06 (2012), P06001. [MQR+] K. Matetski, J. Quastel, and D. Remenik. TASEP and generalizations: Method for exact solution. In preparation. [MQR17] K. Matetski, J. Quastel, and D. Re...

work page Pith review arXiv 2012

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Limiting one-point distribution of periodic TASEP

arXiv: 2008.07024 [math.PR]. [BCF14] A. Borodin, I. Corwin, and P. Ferrari. Free energy ﬂuctuations for directed polymers in random media in 1 + 1 dimension. Comm. Pure Appl. Math. 67.7 (2014), pp. 1129–1214. [BCFV15] A. Borodin, I. Corwin, P. Ferrari, and B. Vet ˝o. Height ﬂuctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18.1 (2015), Art. 20,

work page Pith review arXiv 2014