Convergence of the KMP model to the KPZ equation

Francesco Casini; Guillaume Barraquand

REVIEW 3 major objections 7 minor 2 cited by

Convergence of the KMP model to the KPZ equation

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The KMP energy-transport process, observed in a t^{3/4}-shifted window, converges to the multiplicative-noise stochastic heat equation whose logarithm is the KPZ equation, with noise coefficient 1/(2√α).

desk verdict A credible and significant proof that KMP converges to KPZ in the t^{3/4} window, provided the asserted uniform-in-ε decay bound in Section 6 is supplied. read the letter →

arxiv 2507.19222 v2 pith:FCXN7QOH submitted 2025-07-25 math.PR cond-mat.stat-mechmath-phmath.MP

Guillaume Barraquand , Francesco Casini This is my paper · ORCID

classification math.PRcond-mat.stat-mechmath-phmath.MP MSC 60K3560H15

keywords KMPprocessKPZequationstochasticheatrandomwalkinenvironmentflowofkernelsmoderatedeviationsinteractingparticlesystemsnoisevariance

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 the Kipnis–Marchioro–Presutti (KMP) process, a standard exactly solvable model of energy transport on the integer line, has a scaling limit governed by the KPZ equation. In a window of width $N^{1/2}$ shifted by $N^{3/4}t$, and after an explicit exponential normalization, the energy density field converges as $N \to \infty$ to the multiplicative-noise stochastic heat equation $\partial_t U = \frac{1}{2} \partial_{xx} U + \frac{1}{2\sqrt{\alpha}} U \Xi$, so that $\log U$ solves the KPZ equation. The proof works by identifying KMP energies with the quenched transition kernel of a continuous-time random walk in a space-time random environment, then applying a general convergence theorem for such kernels. If the result is correct, the KMP model belongs to the KPZ universality class in this moderate-deviation regime, with the model-dependent noise strength determined explicitly.

What carries the argument

The load-bearing object is the quenched transition kernel $K_{s,t}(y,x)$ of a continuous-time random walk whose environment is generated by the same Poisson clocks and Beta$(\alpha,\alpha)$ variables that drive the KMP redistribution. Proposition 2.5 identifies the KMP energy at site $x$ with the action of this kernel on an initial localized energy, making the energy field a stochastic flow of kernels in the sense of [SSS14]. That identification converts the SPDE convergence problem into checking six hypotheses from [Par24] and then computing the noise variance, which the paper does through a discrete-time $\varepsilon$-approximation of the kernel flow.

What would settle it

Compute $\gamma_\varepsilon^2$ explicitly for the discrete-time $\varepsilon$-approximation at small $\varepsilon$ and fixed $\alpha$; if it does not approach $1/(4\alpha)$, the claimed noise coefficient fails. Alternatively, exhibit points where the asserted uniform exponential decay of the two-point mixed moments fails, which would break the dominated-convergence step and leave the coefficient unproven.

Watch

Extended reading notes

Core claim

The central result, Theorem 3.1, states that the rescaled field $F_N(t,x) = C_{N,t,x}\,\eta(tN, N^{3/4}t + N^{1/2}x)$, with $C_{N,t,x} = \exp(N^{1/4}x + N^{1/2}t/2 + t/8)$, is tight in $D([0,T], \mathcal{S}'(\mathbb{R}))$, and every limit point coincides with the law of the unique multiplicative-noise stochastic heat equation solution $\partial_t U = \frac{1}{2}\partial_{xx}U + \frac{1}{2\sqrt{\alpha}} U \Xi$ with $U(0,\cdot) = \delta_0$. Equivalently, $\log U$ solves the KPZ equation. This is a moderate-deviation scaling: the observation window sits between the diffusive scale $t^{1/2}$ and the ballistic scale $vt$, where the fluctuations are multiplicative rather than additive. The paper's identification of the KMP dynamics with a stochastic flow of kernels is what turns this statement into a consequence of the general random-walk-in-random-environment convergence theorem.

Load-bearing premise

The proof assumes, without a displayed argument, that there is an exponentially decaying $\ell^1(\mathbb{Z}_{\ge 0})$-valued function $G_{\mathrm{Decay}}$ controlling the two-point mixed moments uniformly in the discretization parameter $\varepsilon$, and it also assumes the external convergence theorem stated as Theorem 3.5; if either premise gives way, the explicit coefficient $1/(2\sqrt{\alpha})$ is not established by this argument.

Editorial extensions

If this is right

The KMP model, not only exclusion-type models, falls in the KPZ universality class at the $t^{3/4}$ moderate-deviation scale, so log-energy fluctuations there obey the KPZ equation.
The noise strength is fixed at $1/(2\sqrt{\alpha})$; any simulation or further theory of KMP fluctuations in this window must reproduce this coefficient.
The proof template applies to any energy redistribution rule that produces a stochastic flow of kernels satisfying the six hypotheses, so the result should extend to multi-site or Beta$(\alpha,\beta)$ redistribution up to constants.
The paper conjectures that at ballistic sites $x \sim vt$ with $v \neq 0$, log-energy has Tracy–Widom fluctuations, connecting the moderate-deviation KPZ regime to the KPZ fixed point.

Reading between the lines

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

A direct numerical check of the two-point covariance in the $\varepsilon$-discretized flow would test the constant $1/(2\sqrt{\alpha})$ even before a proof of the missing uniform decay bound is found.
The missing $G_{\mathrm{Decay}}$ bound is likely obtainable by the same Poisson-event path-counting argument used for hypothesis HP 5; if it fails, convergence might survive with a different noise coefficient, but the theorem as stated would not follow.
The paper's own $\alpha \to 0$ question hints that the limiting kernel flow should be a sticky Brownian flow, which would place the KMP model in the same family as discrete beta walks at weak noise.

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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 KMP-to-KPZ convergence is real, and the proof strategy is the right one. The main new idea is Proposition 2.5: the KMP energy field is exactly the quenched transition kernel of a continuous-time random walk in a space-time Beta environment. That observation turns an interacting particle system into a stochastic flow of kernels and lets the authors invoke Parekh's recent convergence theorem for moderate deviations of RWREs. The resulting Theorem 3.1, with the explicit noise coefficient 1/(2√α), is a genuinely new result: earlier work only had Edwards-Wilkinson limits in the diffusive window or MFT predictions for large deviations. The paper also gives a clean derivation of the k-point motion and duality from the flow picture, which is a nice bonus.

The verification of Parekh's hypotheses is mostly careful. HP1–HP4 and HP6 are handled explicitly, and the one-point annealed kernel is computed exactly. The delicate part is HP5, where the proof is a sketched collision argument with exponential decay asserted rather than written in detail. It reads as fillable, but it is currently a gap.

The bigger soft spot is Section 6. Proposition 6.5, which bridges the discrete-time variance γ_ε² to the continuous-time γ², needs a uniform-in-ε exponentially decaying majorant G_Decay to justify dominated convergence over the infinite sum over z. The paper asserts this bound by 'a similar argument to' the already sketchy HP5 proof, and the text contains no actual proof. Without that bound, the limit N_ε→N and D_ε→D is not justified, and the explicit coefficient 1/(2√α) is not established. This is a specific, fillable technical gap, not evidence of a wrong theorem — the authors have the right mechanism, and the constant comes out of a direct computation of one-step covariances once the approximation is justified.

Secondary dependence on the unrefereed preprint [Par24] is real but not circular; the paper's own contribution is reducing KMP to that framework and verifying hypotheses. If Theorem 3.5 or its hypotheses collapse, the main theorem falls, but that is true for any reduction to an external deep result.

Who is this for? Specialists in interacting particle systems and KPZ universality. The result is significant enough to deserve refereeing. I would send it out with a request to supply the missing bounds. The gaps are specific, and the central argument is sound.

Referee Report

3 major / 7 minor

Summary. The manuscript proves that the one-dimensional KMP energy process, started from a single unit of energy, converges in a t^{3/4} moderate-deviation window to the multiplicative-noise stochastic heat equation with noise coefficient 1/(2√α), equivalently the logarithm of the solution solves the KPZ equation. The proof identifies the KMP process with a stochastic flow of kernels of a continuous-time random walk in a Beta space-time random environment (Proposition 2.5), verifies the hypotheses of Parekh's general convergence theorem (Proposition 4.1), passes from the interpolated field to the Skorokhod-space field (Proposition 5.1), and computes the noise variance through a discrete-time approximation (Propositions 6.5 and 6.7).

Significance. If correct, the result places the KMP model in the KPZ universality class in the moderate-deviation scaling and fixes the model-dependent constant explicitly as 1/(2√α). The stochastic-flow identification is exact and elegant, and the variance computation is largely self-contained and explicit. The paper clearly states its dependence on Parekh's theorem, which is not circular. The main reservations are that the explicit noise coefficient rests on an unproved uniform-in-ε decay bound, and that the verification of one hypothesis of the external theorem (HP5) is only sketched; both points are fillable but load-bearing.

major comments (3)

[Section 6.2, Proposition 6.5] The proof of Proposition 6.5 asserts that, uniformly in ε, there exists an exponentially decaying ℓ¹(Z_{≥0}) function G_Decay bounding the difference of the two-point correlation terms, and says this follows by an argument similar to the proof of HP5. No proof of this bound is supplied. This bound is exactly the z-summable majorant needed to pass from the pointwise convergence in Lemma 6.6 to convergence of the infinite sums N_ε and D_ε; without it, dominated convergence fails and the limit γ_ε²→γ² is not justified. Since Proposition 6.7 then identifies γ² with 1/(4α), the explicit coefficient 1/(2√α) in Theorem 3.1 is not established as written. This is a specific, fillable gap rather than evidence of a wrong result.
[Section 4, Proposition 4.1 (HP5)] The verification of HP5 is only sketched. In the k=2 case, the event E_d is not proved to have the claimed exponentially small probability, and the displayed estimate for E[Δ_i²|E_d] divides by P(E_d) without controlling this denominator; the stated bound C₃e^{−C₄|x₁−x₂|} therefore does not follow from the Cauchy–Schwarz and finiteness-of-Poisson-events remarks as written. The extension to k=3,4 is asserted after 'repeatedly applying the Cauchy-Schwartz inequality' without details. Since HP5 is a hypothesis of Theorem 3.5, the proof of the proposition must be completed.
[Section 3.5, Theorem 3.5] The main theorem inherits its convergence input entirely from the unrefereed preprint [Par24], stated here as Theorem 3.5. This is not circular, and the authors state the dependence clearly, but it is load-bearing: if [Par24, Theorem 1.4] is not correct or not accepted, the arguments in this manuscript do not independently establish Theorem 3.1. The authors should either cite a published or otherwise refereed version, include a self-contained proof of the needed special case, or make the conditional nature of the dependence explicit in the statement of the main theorem.

minor comments (7)

[Section 3.1] In the mild solution formula for the stochastic heat equation, the heat kernel in the noise integral should be p_{t−s}(x−y), not p_t(x−y).
[Section 5] In the display for Aldous's criterion, a minus sign is missing: the probability should be P(|⟨F_N(τ+θ,·),φ⟩ − ⟨F_N(τ,·),φ⟩|>ε).
[Section 6.1, Definition 6.2] The probability for the event that both indicators are zero is (1−ε)², not 1−2ε(1−ε); the displayed expression 1−2ε(1−ε) is the probability that the two indicators are equal.
[Section 6.1, equation (6.3)] The notation p^{(2),ε}_{ε,n} contains a stray subscript ε; it should presumably be p^{(2),ε}_n.
[Section 5, Lemma 5.2] The term o(N) in estimate (5.1) and in the subsequent display should be o(1), since the estimate is meant to vanish as N→∞.
[Section 3.5, equation (3.13)] There are typographical slips in the denominator of γ², where 'p_dif(z.a)' should read p_dif(z,a), and the inner sum notation is otherwise inconsistent.
[Section 4] The sentence 'by the fact that the there can only be a finite number of Poisson event [0,1], we have obtain' contains grammatical errors and should be rewritten.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the KMP-to-KPZ limit reduces to an external theorem [Par24] and to an explicit variance computation; self-citations are background only.

full rationale

Walking the derivation chain: Proposition 2.5 identifies KMP energies with the quenched transition kernel of the RWRE by the same Poisson/Beta dynamics (2.3)-(2.4), an exact algebraic identity rather than an assumption of the target result. Theorem 3.1 is then obtained by checking Assumptions 1 (Proposition 4.1) and applying Theorem 3.5, a result imported from S. Parekh's preprint [Par24]; the authors' role is to verify the hypotheses and identify the model-dependent constants. The computation of gamma^2 = 1/(4 alpha) in Section 6 is self-contained: it defines a discrete-time Bernoulli approximation, proves pointwise convergence of p^(2),epsilon to p^(2) (Lemma 6.6), asserts a uniform exponentially decaying majorant G_Decay to justify dominated convergence (Proposition 6.5), and then evaluates the one-step covariance of the discretized walk (Proposition 6.7). The asserted uniform bound G_Decay is unproved in the text ('we obtain that there exists... uniformly in epsilon'), which is a correctness gap at a load-bearing point, but it is not circular: it is a missing uniform integrability estimate, not a concealed use of the conclusion or of a fitted parameter. Self-citations ([BC17], [BLD20], [BR20]) supply background, analogues, or conjectural context (Conjecture 3.2) and none is load-bearing for Theorem 3.1. The dependence on [Par24] is real but external and not authored by Barraquand/Casini, so by rule 4 it counts as independent support rather than circularity. No step reduces to its own input.

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

No free parameters are fitted to data: α is the fixed, pre-existing parameter of the Beta(α,α) redistribution in the KMP model, and N is a scaling parameter. The main external input is the theorem of [Par24] (stated as Theorem 3.5), an independent preprint that is not verified inside this paper. The paper's own contributions are the exact stochastic-flow identification (Proposition 2.5), the verification of the hypotheses, and the noise variance computation; the latter relies on two sketched decay bounds (the HP5 bound and the uniform-in-ε G_Decay bound) that should be completed. No new physical entities are introduced.

assumptions (5)

domain assumption Correctness of [Par24, Theorem 1.4] in the special case p=1, stated here as Theorem 3.5.
The main convergence statement is imported directly from an independent recent preprint (arXiv:2401.06073); its proof is not reproduced.
domain assumption Existence and uniqueness of the solution to the multiplicative SHE (3.2) in the appropriate class.
Invoked for the definition of the limiting object; the paper cites [BC95, Par19].
ad hoc to paper Exponential decay of the collision probability used in the HP5 verification (C3 e^{-C4|x1-x2|} bound in Proposition 4.1).
The argument is sketched using the event E_d and Cauchy-Schwarz; the explicit exponential decay is asserted rather than proved in detail.
ad hoc to paper Uniform-in-ε exponentially decaying bound G_Decay in the proof of Proposition 6.5.
Asserted ('we obtain') and load-bearing for the dominated convergence that identifies the noise variance; no proof is supplied.
standard math Standard results: Donsker invariance principle, Aldous tightness criterion, Mitoma criterion, Portmanteau theorem.
These are standard tools invoked in Sections 5 and 6; no novelty claimed.

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Pith. "Pith review of Convergence of the KMP model to the KPZ equation." pith.science (2026). https://pith.science/paper/FCXN7QOH

@misc{pith2026250719222,
  author       = {Pith},
  title        = {Pith review of: Convergence of the KMP model to the KPZ equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCXN7QOH}},
  note         = {Machine review of arXiv:2507.19222}
}

abstract

We prove that the Kipnis-Marchioro-Presutti (KMP) process converges to the Kardar-Parisi-Zhang (KPZ) equation, as time $t$ goes to infinity, in a properly scaled observation window shifted by $t^{3/4}$. Our proof is based on identifying the KMP process with a stochastic flow of kernels describing transition probabilities in a certain model of random walk in space-time random environment. This allows to apply a recent result of arXiv:2401.06073 proving convergence of the density field of random walks in random environment to the KPZ equation in a suitably general sense.

Figures

Figures reproduced from arXiv: 2507.19222 by the authors.

**Figure 1.** The horizontal segments (bridges) represent the times of the Poisson point process attached to each bond. They are decorated with random variables B ∼ Beta(α, α). The continuous red line is a possible trajectory of random walk, from (0, x) to (τ, x − 1). The gray circles represent the location of the particle after each Poisson event. The dynamic of the RWRE (X(t))t>0, yield dynamic for its transition kernel. For so… view at source ↗

**Figure 2.** Diagram in space time coordinates showing the expected scaling limits of the KMP model with initial condition η(0, x) = 1x=0. In the gray area, the energy field is frozen (equal to zero). In the diffusive scaling window in the middle, the KMP model is expected to converge to the stochastic heat equation (based on [BRAS06, Yu16]). Conjecture 3.2 is about the scaling x = vt, v ∈ (0, 1), in green. Our main result (The… view at source ↗

**Figure 3.** Brick-wall KMP model on a segment of length 8 between times n = 1 and n = 6. The vertical lines represent the energy η(n, x) flowing upward in time (for simplicity of the picture, we only report the initial and the final energies). The boxes represent the redistribution mechanism, depending on a Beta variable, between sites x and x+ 1 at time n and according to the recursion relation (A.4). The reflecting boundary c… view at source ↗

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Forward citations

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The averaging process has KPZ critical scale λ=7/8 rather than the moment-criterion value 3/4, with tilted fields converging to the multiplicative SHE of noise 1/√2.
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For random walks in a Dirichlet random environment, rare-trajectory fluctuations match KPZ exponents in d=1,2, and in d=3 an exact second-moment calculation bounds the disorder transition at v_c ≥ 0.639.

Reference graph

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12 extracted references · 10 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Rizkallah, A

[RGIB23] P. Rizkallah, A. Grabsch, P. Illien, and O. Bénicho u. Duality relations in single-ﬁle diﬀusion. Journal of Statistical Mechanics: Theory and Experiment , 2023(1):013202,

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Zarfaty and B

[ZM16] L. Zarfaty and B. Meerson. Statistics of large curren ts in the Kipnis–Marchioro–Presutti model in a ring geometry. Journal of Statistical Mechanics: Theory and Experiment , 2016(3):033304,

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Bettelheim and B

[BM24] E. Bettelheim and B. Meerson. Full statistics of regu larized local energy density in a freely ex- panding Kipnis–Marchioro–Presutti gas. Journal of Statistical Mechanics: Theory and Experiment , 2024(11):113204,

work page 2024

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At the edge of a cloud of Brownian particles

[BW22] D. Brockington and J. Warren. At the edge of a cloud of B rownian particles. arXiv preprint arXiv:2208.11952,

work page Pith review arXiv

[5] [2004]

On the notion(s) of duality for Markov processes

[JK12] S. Jansen and N. Kurt. On the notion (s) of duality for M arkov processes. arXiv:1210.7193,

work page Pith review arXiv

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[Kar94] M. Kardar. Directed paths in random media. arXiv preprint cond-mat/9411022 ,

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[TW92] C. A. Tracy and H. Widom. Level-spacing distribution s and the airy kernel. Commun.Math.Phys. 159 (1994) 151-174 , 159(1):151–174, January

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