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Large time cumulants of the KPZ equation on an interval

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Pith's one-line read A single functional equation determines every large-time cumulant of the open KPZ height.

desk verdict First explicit formulas for all large-time cumulants of the open KPZ equation on an interval, cross-checked through c2 by two independent methods; c3 and above rest on a clearly flagged but unproved exchange of limits. read the letter →

arxiv 2504.18292 v3 pith:MM64FDKX submitted 2025-04-25 math-ph cond-mat.stat-mechmath.MPmath.PR

Guillaume Barraquand , Pierre Le Doussal This is my paper · ORCID
classification math-phcond-mat.stat-mechmath.MPmath.PR MSC 60H1582B2382C22
keywords openKPZequationlarge-timecumulantsfunctionalreplicaBetheansatzASEPfixedpointonanintervaluppertaillargedeviationscumulantgeneratingfunction
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 claims that for the Kardar-Parisi-Zhang equation on an interval with Neumann boundary parameters $u,v$, every height cumulant (connected moment) has a large-time limit $\kappa_k(h(x,t)) \sim c_k(L,u,v)\, t$, and that all coefficients $c_k(L,u,v)$ are computable from one parametric functional equation. In that equation, the cumulant generating function $E(s)$ is fixed by $s = \int_{i\mathbb{R}} \frac{dw}{2i\pi} U(w)$ and $E(s) = -s/24 + \frac12\int_{i\mathbb{R}} \frac{dw}{2i\pi} w^2 U(w)$, where $U(w)$ solves $U(w) = -\frac12\log\bigl(1 - 2\zeta \Psi(w) e^{kU}(w)\bigr)$; eliminating the free parameter $\zeta$ between the two integrals yields $E(s)$, whose Taylor coefficients are the $c_k$. The authors reach this system by two independent routes, a scaling limit of open-ASEP cumulants and a replica Bethe ansatz adapted from Brunet and Derrida's periodic computation, and they check agreement for the first two cumulants. If correct, this yields the first exact computation of all large-time cumulants, of the upper-tail large deviation rate function, and of the corresponding fixed-point cumulants in the large-$L$ scaling $u=\tilde u/\sqrt{L}$, $v=\tilde v/\sqrt{L}$.

What carries the argument

The load-bearing object is the pair $(\Psi, k)$: $\Psi(w)$ is the meromorphic function that already normalizes the stationary measure of the open KPZ equation, and $k$ is an integral operator with kernel $-2(\psi(1+w-w')+\psi(1+w'-w))$, built from the digamma function (the logarithmic derivative of the Gamma function). The functional equation $U = -\frac12\log(1-2\zeta\Psi e^{kU})$ is the central mechanism; expanding it in $\zeta$ produces a tree expansion in which every coefficient $U_n$ is a sum of terms $\Psi^c \prod_j k(F_j)$, and the same expansion, after reversion of the series $s(\zeta)$, yields every cumulant $c_k$. The simplification in the large-$L$ limit is that $k$ becomes subleading, so the equation becomes local in $w$ and the fixed-point cumulants are determined by the single profile $\phi(y)$.

What would settle it

Simulate the stochastic heat equation on $[0,1]$ with $u=v=1$, estimate $\kappa_3(h(0,t))/t$ for large $t$, and compare with the closed-form $c_3$ from (3.52); a persistent disagreement beyond numerical error would invalidate the analytic continuation at $k=3$.

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Extended reading notes

Core claim

The central discovery is that the asymptotic cumulant generating function of the open KPZ height is carried by the functional equation $U(w) = -\frac12\log(1 - 2\zeta \Psi(w) e^{kU}(w))$, with $\Psi(w) = \frac{\Gamma(u+w)\Gamma(u-w)\Gamma(v+w)\Gamma(v-w) e^{w^2L}}{\Gamma(2w)\Gamma(-2w)}$ and $k$ the convolution operator whose kernel is $-2(\psi(1+w-w')+\psi(1+w'-w))$. Once $U$ is expanded in powers of $\zeta$, the integrals $s=\int_{i\mathbb{R}} \frac{dw}{2i\pi} U(w)$ and $E=-s/24+\frac12\int_{i\mathbb{R}} \frac{dw}{2i\pi}w^2 U(w)$ give $s$ and $E$ as power series in $\zeta$; eliminating $\zeta$ and re-expanding in $s$ produces the cumulants through the triangular recurrence (1.38). The same framework gives the first cumulant as $c_1=-1/24+\frac12\partial_L\log Z_{u,v}(L)$ and the second as an integral against the stationary-measure normalizing density $\nu(dw)$, and in the large-$L$ limit it reduces to a local equation whose coefficients are elementary integrals of $\phi(y)=\frac{4y^2 e^{-y^2}}{(\tilde u^2+y^2)(\tilde v^2+y^2)}$, characterizing the two-parameter KPZ fixed point on an interval.

Load-bearing premise

The load-bearing premise is that the particle-model-to-PDE convergence can be exchanged with the large-time cumulant limit (equivalently, that replica energies known for positive integers continue analytically to real parameters), a step verified only for the first two cumulants.

Editorial extensions

If this is right

  • Every $c_k(L,u,v)$ can be computed iteratively from the coefficients $U_n$; the paper displays explicit $c_1,c_2,c_3$, and the recurrence (1.38) gives the rest.
  • The upper-tail probability obeys $P(h(0,t)/t + 1/24 > H) \sim e^{-t\Phi(H)}$, with $\Phi$ obtained by Legendre transform of $E(s)+s/24$.
  • In the scaling $u=\tilde u/\sqrt{L}$, $v=\tilde v/\sqrt{L}$, the cumulants scale as $c_k \sim L^{(k-3)/2}\tilde c_k$, and the $\tilde c_k$ are closed-form integrals in terms of $\phi(y)$; this is the large-time characterization of the two-parameter KPZ fixed point on an interval.
  • The periodic KPZ equation is recovered by replacing $\Psi(w)$ with $e^{Lw^2/2}$, and the paper verifies that the second cumulant agrees with the Brunet-Derrida result.
  • For general boundary parameters with $u,v>0$, $c_1$ is tied to the stationary-measure normalization through $c_1=-1/24+\frac12\partial_L\log Z_{u,v}(L)$, and analytic continuation covers $u\leq 0$ or $v\leq 0$.

Reading between the lines

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

  • The same functional equation may control the crossover regime $t \sim L^{3/2}$, not only the limits $t\gg L^{3/2}$ and $L\to\infty$; the paper does not derive finite-time formulas, so this is an extrapolation.
  • Because $c_2$ is expressed as an $L$-derivative of a kernel average, the higher $c_k$ might also be total $L$-derivatives of tree functionals of the stationary measure, which would make them directly accessible to Monte Carlo sampling of the measure (1.5).
  • If the assumed commutation of limits holds, the same ASEP-to-KPZ scaling should yield the interval cumulants of other integrable stochastic models in the KPZ class, such as the stochastic six-vertex model, a route the paper mentions but does not take.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies the KPZ equation on an interval [0,L] with Neumann boundary parameters u,v. The main claim is that, as t tends to infinity, the k-th cumulant of the height is κ_k(h(0,t)) = c_k(u,v,L)t + o(t), and that the cumulant generating function E(s) is given parametrically by (1.14)–(1.16): s = ∫ U, E = -s/24 + (1/2)∫ w^2 U, where U solves U(w) = -(1/2)log(1 - 2ζΨ(w)e^{kU}(w)), with Ψ(w) given by (1.17) and the operator k by (1.18)–(1.19). The authors give the first few cumulants explicitly, analyze the large-L scaling limit u = ũ/√L, v = ṽ/√L leading to a two-parameter KPZ fixed point, and use a Legendre transform to write an upper-tail rate function. Two methods are used: a weakly asymmetric scaling limit of open ASEP (Section 2) and a replica Bethe ansatz adapted from Brunet and Derrida (Section 5). The paper is explicitly heuristic: Section 1.4 lists the unproved steps, including the replica analytic continuation in the number of replicas, the exchange of large-time and scaling limits, and the selection of the ground-state branch among solutions of the functional Bethe equation.

Significance. If the formulas are correct, this is an important advance: it gives the first exact large-time cumulants for the open KPZ equation with general boundary parameters, including higher cumulants and a conjectural KPZ fixed point rate function. The agreement of the two methods for c1 and c2, the recovery of the periodic Brunet–Derrida results and of open-TASEP formulas in the large-L limit, and the explicit checks in special cases are significant strengths. The manuscript is also unusually candid about its lack of mathematical proof. However, the independent verification stops at order two in the replica expansion, so the higher cumulants, which are the paper's main new content, remain conditional on the heuristic steps.

major comments (3)
  1. [§2.4 and §5.4] The formulas for c_k with k ≥ 3 are not independently supported. Section 2.4 obtains E(s) by taking the weak-asymmetry limit ε → 0 in the ASEP cumulant generating function, but the convergence theorem [38,39] controls typical fluctuations rather than the exponential tilts entering E(µ); Section 1.4 explicitly acknowledges this. The replica method provides an independent route, but it is carried out only to order n^1 in the expansion of M(z): c1 and c2 are derived in (5.49) and (5.66), and no replica computation of c3 is given. Since c3 is the first order at which the two methods could disagree if the exchange of limits fails, the central claim that all c_k are given by (1.14)–(1.16) is not yet established for k ≥ 3. A concrete remedy is to extend the replica computation to order n^2 and compare with (3.52).
  2. [§6, Eqs. (6.1)–(6.4)] There appears to be a factor-of-two inconsistency in the periodic-case reduction. If Uper = (1/2)U|_{Ψ=e^{L/2w^2}} as stated after (6.1), then substituting into (1.16) gives Uper = -(1/4)log(1 - 2ζΨ e^{2kUper}), not (6.1); conversely, the solution of (6.1) does not satisfy (1.16) under this identification. Moreover, combining (6.2)–(6.3) with U1 = Ψ' gives c1per = -1/24 - 1/(4L) (since ∫ν w^2 = -1/(2L)), whereas (6.6) states -1/24 - 1/(2L). Either (6.1), the identification Uper = U/2, or the relation (6.4) needs correction, and the claimed agreement with [16, Eq. (49)] should be rechecked.
  3. [§1.2.3 and §1.3.1] The upper-tail large deviation statement (1.10)–(1.13) presupposes that the parametric representation (1.14)–(1.16) defines E(s) for all real s and that E(s) + s/24 is convex. The paper computes E(s) only as a power series around s = 0 (Section 1.2.3) and does not establish convergence or a global definition; the Legendre transform (1.13) therefore does not yet yield a rate function on the whole upper tail. A statement of the domain of s on which (1.14)–(1.16) is valid, or a direct large-deviation argument, is needed.
minor comments (5)
  1. [Abstract] In the abstract, the sentence 'u = ũ/√L, u = ṽ/√L' should read 'v = ṽ/√L'.
  2. [Eq. (3.42)] Equation (3.42) has unbalanced parentheses in the denominator of the last integral; the intended normalization is unclear as printed.
  3. [§3.4.2] In Section 3.4.2, 'we already now that f(u,-u) = u^2' should be 'we already know'.
  4. [Remark 2.3] Remark 2.3 says 'for u = 0, v = 1/2 one has Ψ(u) = 4π e^{w^2L}'; the argument should be Ψ(w), not Ψ(u).
  5. [Footnote 1 and §1.2.1] There are two spelling typos: 'Stritly speaking' in footnote 1 and 'Brownien motion' in Section 1.2.1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the cumulant formulas are derived from independent ASEP and replica inputs, with only an explicitly flagged non-circular exchange-of-limits assumption for k>=3.

full rationale

No load-bearing step in this paper reduces by construction to its own input. The central system (1.14)-(1.16) is obtained in Section 2 as a scaling limit of the exact ASEP current large-deviation functional (2.17)-(2.19), which is taken from external sources [20,21] and is not fitted to the target KPZ cumulants; the same functional equation is reached independently in Section 5 from a replica Bethe-ansatz calculation of E0(n,L), with c1 and c2 matched in (5.49) and (5.66). No parameter is fitted to any c_k: zeta is an auxiliary variable eliminated between (1.14) and (1.15), and the c_k are then read off by the triangular recurrence (3.6). The self-citations [31,37] are not load-bearing: c1 is rederived at (3.35) and (5.49), and the stationary-measure scaling of [31] is used only for the conjectural large-L fixed-point limit, which the paper explicitly labels as expected rather than proved. The strongest caveat is the unproved commutation of the ASEP-to-KPZ weak convergence [38,39] with large-time exponential tilts, flagged in Section 1.4 and in the Introduction; the authors check only c1 and c2. That is an assumption about limit exchange, not a circular reduction: if it failed, c_k for k>=3 would be incorrect, but they would not be equal to the inputs by construction. I therefore assign score 1.

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

No numerical data are fitted: the parameter zeta is eliminated and all constants are mathematical. The central claim imports from prior literature the ASEP large-time current functional equations [20,21] and the stationary normalization of the open KPZ equation [30,31,32,37]; these are domain assumptions. The main ad hoc postulates are the replica analytic continuation, the exchange of limits, and the choice of ground-state branch.

assumptions (5)
  • domain assumption Completeness of Bethe ansatz eigenfunctions for the Lieb-Liniger Hamiltonian H_n on the interval.
    Sections 1.4 and 5.1: the ground state energy E0(n,L) is extracted from Bethe ansatz; completeness is needed for the spectral representation of E[Z^n].
  • ad hoc to paper Analytic continuation in n from positive integers to real s (replica trick).
    Sections 1.4 and 5.2: the cumulant generating function is obtained by interpolating E0(n,L) from integer n; uniqueness of this continuation is assumed.
  • ad hoc to paper Exchange of large-time and ASEP-to-KPZ scaling limits for large deviations.
    Sections 1.4 and 2.4: convergence of ASEP to KPZ is known for typical fluctuations; the paper assumes the large-time cumulant generating function also commutes with the limit, and verifies this only for c1 and c2.
  • ad hoc to paper Selection of the ground-state branch among solutions of the functional Bethe equation.
    Section 5.2 and Remark 5.4: the function g is chosen with smallest growth at infinity; other choices are assumed to correspond to excited states.
  • standard math The power series in zeta and in s have positive radius of convergence.
    Sections 3.1 and 4.2: cumulants are extracted by series inversion; convergence is assumed, with a finiteness bound stated for the coefficients phi_k and psi_k in Section 4.2.

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Pith. "Pith review of Large time cumulants of the KPZ equation on an interval." pith.science (2026). https://pith.science/paper/MM64FDKX

@misc{pith2026250418292,
  author       = {Pith},
  title        = {Pith review of: Large time cumulants of the KPZ equation on an interval},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MM64FDKX}},
  note         = {Machine review of arXiv:2504.18292}
}
abstract

We consider the Kardar-Parisi-Zhang equation on the interval $[0,L]$ with Neumann type boundary conditions and boundary parameters $u,v$. We show that the $k$-th order cumulant of the height behaves as $c_k(L,u,v)\, t$ in the large time limit $t \to +\infty$, and we compute the coefficients $c_k(L,u,v)$. We obtain an expression for the upper tail large deviation function of the height. We also consider the limit of large $L$, with $u=\tilde u/\sqrt{L}$, $u=\tilde v/\sqrt{L}$, which should give the same quantities for the two parameter family $(\tilde u, \tilde v)$ KPZ fixed point on the interval. We employ two complementary methods. On the one hand we adapt to the interval the replica Bethe ansatz method pioneered by Brunet and Derrida for the periodic case. On the other hand, we perform a scaling limit using previous results available for the open ASEP. The latter method allows to express the cumulants of the KPZ equation in terms a functional equation involving an integral operator.

Figures

Figures reproduced from arXiv: 2504.18292 by the authors.

Figure 1
Figure 1. Graph of the function u 7→ c2(u, u, 1). which is a simple consequence of (1.23) and (1.24), we see that the second cumulant can also be written as a total derivative (with respect to L) of a simpler expression: c2(u, v, L) = 1 2 ∂L Z iR ν(dw1) Z iR ν(dw2) ¯k(w1, w2)  (1.28) where we have defined the reduced kernel ¯k(w1, w2) = k(w1, w2) + δ(w1 − w2). (1.29) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. We consider a function F defined for u > 0 by an integral of the form F(u) = R iR f(u, w) dw 2iπ where w 7→ f(u, w) is a meromorphic function with poles at w = u and w = −u, and u 7→ f(u, w) is also holomorphic except at w = ±u. Then, its analytic continuation to u < 0 is R C f(u, w) dw 2iπ where C is the contour shown above. This contour is such that u still lies on the right of the contour (as is the case when u >… view at source ↗

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    (C.10) Hence our calculation using the kernel K gives the same function M(z) as the one of of [16] at least to the order O(n2) in the ground state energy. It shows, among other things, that the way we have fixed some arbitrariness in the intermediate steps of the calculation i...

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Reviewed August 16, 2026 · model on record in the stance chip above.