Derrida-Retaux type models and related scaling limit theorems

Run Zhang; Zenghu Li

REVIEW 1 major objections 6 minor 1 cited by

Derrida-Retaux type models and related scaling limit theorems

T0 review · 1 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that generalized Derrida–Retaux recursions, rescaled in time by k and space by 1/k, converge to a unique continuous-time Markov process characterized by an explicit generator, a martingale problem, and a Poisson-driven…

desk verdict Solid scaling-limit theorems for generalized Derrida–Retaux models, but the headline process-level theorem needs an extra finite second-moment condition on the offspring law. read the letter →

arxiv 2411.12189 v2 pith:EI4VZN6H submitted 2024-11-19 math.PR

Zenghu Li , Run Zhang This is my paper · ORCID

classification math.PR MSC 60H2060J2560J76

keywords Max-typerecursivemodelDerrida–RetauxtransitionsemigroupgeneratormartingaleproblemweakconvergenceSkorokhodspacescalinglimit

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 generalized Derrida–Retaux recursion — a discrete scheme in which a value is replaced by the positive part of itself plus an offspring-distributed jump minus 1 — has a genuine continuous-time limit, and it characterizes that limit completely. With time rescaled by k, space by 1/k, and the renewal rate set to a/k, the marginals converge in Wasserstein distance to the unique solution of the differential equation (2.9) at the explicit rate $W(\gamma^{(k)}_{\lfloor kt\rfloor},\mu_t)\le e^{a(m_1+2)t}[\frac{4}{k}(1+at)+W(\gamma^{(k)}_0,\mu_0)]$, and the whole rescaled paths converge weakly in Skorokhod space to a Markov process. That process is described in four equivalent ways: a transition semigroup with generator $A_t$ from (1.8), a martingale problem, an SDE driven by a Poisson random measure (1.10), and the closed entrance law $\mu_t$ for the semigroup. This matters because the continuous-time model is the regime in which the Derrida–Retaux phase transition — pinning, free-energy asymptotics, and the predicted $4/n^2$ sustainability probability — is exactly solvable, and the theorem supplies the rigorous bridge from the discrete recursions to that regime.

What carries the argument

The engine of the paper is the pair formed by the generator $A_t$ and the Wasserstein distance built on the truncated metric $\rho(x,y)=1\wedge|x-y|$. The generator, $A_t f(x)=a\int_{\mathbb R_+}[f(x+z)-f(x)]\mu_t^q(dz)-f'(x)\mathbf 1_{\{x>0\}}$, has two parts: a pure-jump term that at rate $a$ adds a random amount distributed as $\mu_t^q$, the offspring mixture $\sum_k q_k\mu_t^{*k}$ of the current marginal, and a drift of $-1$ that operates only while the process is positive. The stochastic equation (1.10), driven by a Poisson random measure with intensity $a\,ds\,du$ and with $G_s^{-1}$ the right-continuous inverse of $\mu_s^q$, is the pathwise realization of the same generator, and Theorem 3.2 shows the martingale problem and the SDE describe the same process. Quantitatively, the dual representation (2.2) and the convolution inequality of Lemma 2.2 convert the recursive equation into an integral inequality; Gronwall's inequality then yields both the contraction $W(\mu_t,\gamma_t)\le e^{am_1 t}W(\mu_0,\gamma_0)$ between any two solutions and the explicit scaling error (2.19). Existence is produced by a successive-approximation iteration over sub-probabilities (2.14), and path tightness in Lemma 5.4 uses the stopping-time tightness criterion together with the moment bounds of Lemmas 5.1–5.3.

What would settle it

Run the rescaled recursion (2.18) with an infinite-mean offspring law such as $q_k=c k^{-3/2}$: if the marginals $\gamma^{(k)}_{\lfloor kt\rfloor}$ still converge for every $t$ to a solution of (2.9), the finite-mean premise is stronger than needed, while a failure to converge (or several limit points) would show the premise is load-bearing. As a second check, in the solvable case $a=1$, $q_1=1$ with an exponential-type initial law, the simulated marginals should match the closed-form CDR flow with error of order $1/k$; a persistent mismatch would point to a concrete error in Theorem 2.7 or Theorem 5.5.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is that the generalized discrete Derrida–Retaux dynamics (1.6), after the rescaling $\gamma^{(k)}_n(dx)=\mu^{(k)}_n(k\,dx)$ with renewal rate $\alpha=a/k$, converge as $k\to\infty$ to the generalized CDR model: the unique family of probability measures solving $\partial_t\langle\mu_t,f\rangle=a\langle\mu_t*\mu_t^q-\mu_t,f\rangle-\langle\mu_t,f'\mathbf 1_{(0,\infty)}\rangle$, equation (2.9). Theorem 2.7 makes this quantitative with the explicit Wasserstein bound above, and Theorem 5.5 lifts it to the process level: whenever the rescaled initial laws converge weakly to $\mu_0$ and satisfy the second-moment condition (5.2), the rescaled chains $(Y^{(k)}_{\lfloor kt\rfloor}:t\ge 0)$ converge in $D([0,\infty),\mathbb R_+)$ to the generalized CDR process $X_t$ with initial law $\mu_0$, the pathwise unique solution of the stochastic equation (1.10). Along the way the paper proves the process is equivalently a Markov process with inhomogeneous transition semigroup $(P_{r,t})$ generated by $A_t$ in (1.8), a solution of the $(A_t)$-martingale problem, and a weak solution of (1.10), with $\mu_t$ the closed entrance law of the semigroup.

Load-bearing premise

The argument stands on the offspring distribution having finite mean, $\sum_{k\ge1}kq_k<\infty$, which buys existence, uniqueness, contraction, and both limit theorems, and the path-level theorem additionally assumes the rescaled initial laws have uniformly bounded second moments.

Editorial extensions

If this is right

The rescaled discrete chain becomes a computable approximation of the continuous-time model: once the rescaled starting law is close to $\mu_0$, the bound (2.19) keeps the Wasserstein error after $k$ time steps at $O(1/k)$ times a factor exponential in $t$, and Theorem 5.5 lifts the approximation from marginals to whole paths in the Skorokhod space.
For every offspring law with finite mean, the limiting process is one well-characterized object: the equivalence of the transition semigroup, the martingale problem, and the SDE (Theorems 3.2 and 4.7) means a result proved in any one representation automatically applies to the others.
The path-space limit transfers convergence to path functionals, so hitting probabilities, excursion statistics, and the quantities entering the pinning–unpinning phase transition of the discrete models converge to those of the continuous-time process.
The moment recursions (5.6) give explicit growth rates — $e^{am_1t}$ for the mean and $e^{a(2m_2+1)t}$ for the mean square — so the qualitative growth of the discrete model is governed by the first two moments of the offspring law exactly as for the limit process.

Reading between the lines

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

Extension the paper does not state: the $1/k$ rate in (2.19) should be the leading-order average error, and a second-order expansion of the rescaled chains around the CDR path would presumably yield a $k^{-1/2}$ Gaussian fluctuation; proving that would upgrade convergence to a distributional approximation usable for error bars in simulation.
Because the generator (1.8) depends on the offspring law only through the mixture flow $\mu_t^q$, two laws that generate the same mixture flow would produce identical CDR processes — a reducibility the paper does not exploit and that could simplify simulation by replacing a complicated $q$ with a lighter one.
The second-moment premise (5.2) is used only for tightness, so the process-level theorem may well survive under a first-moment condition; testing initial laws with $E[(X_0^{(k)})^2]\sim k^{2+\varepsilon}$, so that (5.2) fails but the rescaled states still converge weakly, would show whether the premise is sharp.
A renormalization reading: iterating the (time $k$, space $1/k$) rescaling composes the discrete dynamics, and the CDR semigroup is the continuous-time flow of that composition; the critical pinning regime would then correspond to a nontrivial fixed point, and the solvable continuous-time setting is a plausible arena for extending the free-energy asymptotics (1.2) to general offspring laws.

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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

Quick take: this is a genuinely useful paper. It makes rigorous the informal scaling from Derrida–Retaux and Hu–Mallein–Pain by proving Wasserstein convergence of the rescaled discrete dynamics to a generalized CDR equation and then going further: a martingale problem, an SDE representation, and a Skorokhod-space process-level limit. The main estimate (2.19) is clean, and the characterization theorems (3.2, 4.7) are argued carefully. This is the first place I know of where the process-level limit is proved for general offspring distribution q, not just the exponential case.

The soft spots are real but not fatal. The stress-test note is on target: the tightness proof in Section 5 uses the second moment m2 of the offspring law, and m2 = ∞ is not excluded by the standing assumption m1 < ∞. Lemma 5.1's second-moment bound, Lemma 5.2's E[sup Y^2] finiteness, and Lemma 5.3's quadratic variation estimate all blow up if m2 = ∞. So Theorem 5.5 as stated is proven only under the additional, unstated condition m2 < ∞, or under a different tightness argument that does not rely on finite second moments. That is a missing hypothesis, not a cosmetic gap. The authors should either add m2 < ∞ to Theorem 5.5 or fix the proof.

Also, the introduction says the SDE (1.10) has a pathwise unique solution, but I did not find a proof of that claim anywhere in the body. That is a minor assertion that should be either proved or downgraded to 'weak uniqueness', which is what the martingale problem gives. A few typos and notational slips are scattered around, but nothing that obscures the arguments.

Net: if the moment issue is patched, this is a solid contribution to the recursive-models literature. The Wasserstein bound and the generator characterization will be cited. I would take it for the reading group, and I would send it to a serious referee. My recommendation is to engage with the paper, but condition acceptance of the final version on the Section 5 hypothesis being fixed.

Referee Report

1 major / 6 minor

Summary. The paper studies continuous-time generalized Derrida--Retaux (DR) type models. It introduces a generalized CDR model through equations (1.7) and (2.9), proves existence, uniqueness, and Wasserstein stability under the finite-mean condition m_1<\infty, and shows that rescaled discrete-time generalized DR dynamics converge in Wasserstein distance (Theorems 2.7 and 2.8). It then characterizes the transition semigroup, generator, and martingale problem of the associated Markov process (Theorems 3.2 and 4.7), and proves weak convergence of the rescaled processes in the Skorokhod space D([0,\infty),\mathbb{R}_+) (Theorem 5.5). The proofs are based on explicit integral-equation comparisons, Gronwall's inequality, Itô's formula, Lévy--Khintchine uniqueness, and Aldous-type tightness arguments.

Significance. If the results hold, they provide a rigorous scaling-limit justification for the continuous-time DR equation used in the physics literature and extend the exactly solvable CDR model of Hu--Mallein--Pain to general offspring distributions. The paper's strengths are its explicit quantitative bounds, especially (2.19), the self-contained construction of the transition semigroup, and the clean martingale-problem characterization. The main caveat is that Theorem 5.5 is proved only under an additional second-moment condition on the offspring law that is not stated in the theorem; this is repairable but narrows the theorem's scope as written.

major comments (1)

[Section 5, Lemmas 5.1-5.4 and Theorem 5.5] The process-level limit theorem is proved only under an unstated second-moment condition on the offspring distribution. Lemma 5.1's second estimate (5.6), the martingale-variance bound (5.12), and Lemma 5.3 all require m_2<\infty; without it the right-hand sides are infinite. The standing assumptions before (5.1) and the statement of Theorem 5.5 state only m_1<\infty (from earlier sections) and (5.2). If q has m_1<\infty but m_2=\infty, for example q_j \sim c j^{-3}, and X_0^{(k)}=k so that (5.2) holds, then E[(Y_1^{(k)})^2]=\infty and the estimates (5.6) and (5.12) fail. Thus the tightness argument in Lemma 5.4 does not apply to the hypotheses as stated. This is a missing hypothesis, not a cosmetic issue; please add m_2<\infty to the standing assumptions and Theorem 5.5, or replace the Aldous-type tightness proof by one that avoids second moments of the jump distribution.

minor comments (6)

[Proposition 2.5 proof] In the line following Lemma 2.2, the term '\gamma_t * \gamma^q_s' should read '\gamma_s * \gamma^q_s'; the intended estimate is clear from the context.
[Theorem 3.2 proof] In the equation after 'It follows that', the factor '\mu^a_s(dz)' should be '\mu^q_s(dz)', and 'remains trues' should be 'remains true'.
[Proposition 4.6 proof] The displayed boundary term uses the factor e^{a(r-t)}, which is independent of the integration variable s and would not yield the stated \partial_r formula; it should be the s-dependent atom mass from (4.12). The final backward equation is nevertheless correct, but this displayed step needs repair.
[Lemma 5.4 proof] In the displayed decomposition of Y^{(k)}_{\lfloor k(\tau_k+\delta_k)\rfloor} - Y^{(k)}_{\lfloor k\tau_k\rfloor}, the final martingale term should be M^{(k)}_{\lfloor k\tau_k\rfloor}, not M^{(k)}_{\lfloor k\delta_k\rfloor}; the subsequent estimate uses the \tau_k version.
[Theorem 5.5 statement] The theorem's hypotheses should explicitly include the standing second-moment assumption (5.2), since it is used in Lemma 5.4 and is not visible from the theorem statement alone.
[Abstract and introduction] The abstract and introduction state the scaling-limit result without mentioning the additional m_2<\infty condition needed for Theorem 5.5; once the theorem is amended, these statements should be qualified accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the continuous-time model is defined independently and the discrete-to-continuous convergence is proved by explicit comparison, not assumed.

full rationale

The central claims are the existence and uniqueness of the generalized CDR model (Prop. 2.4-2.6), the Wasserstein convergence of the rescaled discrete dynamics (Thm. 2.7-2.8), and the process-level Skorokhod convergence (Thm. 5.5). None of these reduces to its inputs by construction. Theorem 2.7 takes the continuous solution μ_t of the integral equation (2.12), whose existence is established separately by a Picard iteration in Proposition 2.6, and compares the rescaled discrete recursion (2.18) with that equation directly; the error is controlled by Lemma 2.2 and Gronwall's inequality. The process-level theorem then combines the resulting finite-dimensional convergence with tightness via Aldous's criterion and identifies the limit through the martingale problem (1.9), whose equivalence with the Markov transition semigroup is proved in Theorem 4.7 using the backward equation (4.11). The citations to [6], [9], and [12] motivate or give background but are not load-bearing: the uniqueness of the model is reproved under m1<∞, and [12] is only mentioned for a different asymptotic result. The only notable issue is that Lemmas 5.1-5.3 and 5.4 use the second moment m2 of q, while the theorem's hypotheses do not state m2<∞; this is a missing-assumption validity gap, not a circularity, because the estimates are not assumed as the conclusion.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

No fitted parameters exist in the paper. The model inputs a and q are external; m1 and m2 are derived from q. Standard probabilistic tools are invoked. The generalized CDR process is a defined mathematical object, not an unexplained entity; it is shown to be the unique limit and to satisfy explicit equations, so independent evidence is the theorem itself.

assumptions (10)

standard math Wasserstein duality representation for the ρ-Wasserstein distance
Used in Lemma 2.1 and throughout Section 2.4 to convert measure estimates into Wasserstein bounds; cited to Chen [2].
standard math Gronwall's inequality
Used to obtain the exponential bounds in Propositions 2.5 and Theorem 2.7.
standard math Uniqueness of the Lévy-Khintchine representation
Used in Theorem 3.2 to identify the compensator of the jump measure and hence the Poisson representation.
standard math Aldous's tightness criterion in D([0,∞), R+)
Used in Lemma 5.4 to prove tightness of the rescaled processes.
standard math Martingale representation theorem for point processes
Used in Theorem 3.2 to reconstruct the Poisson random measure N from the compensated jump measure; cited to Ikeda-Watanabe [11].
standard math Itô's formula for càdlàg semimartingales
Used in Theorem 3.2 to pass between the process X, its exponential Z, and the SDE representation.
domain assumption Finite mean offspring: m1 = Σ k q_k < ∞
Assumed from equation (2.3) onward; needed for existence (Prop 2.6), contraction (Prop 2.5), the Wasserstein convergence bound (Theorem 2.7), and moment estimates (Lemma 5.1).
domain assumption Fixed offspring distribution q under rescaling
The paper scales α = a/k and divides space by k, but q is unchanged; this is why the limit equation uses µ^q. It is the natural scaling but is an explicit modeling choice.
domain assumption Second moment condition on rescaled initial laws (5.2)
sup_k k^{-2} E[(X_0^{(k)})^2] < ∞ is assumed before Lemma 5.1 and is used for tightness; without it the process-level theorem is not proved.
domain assumption Filtration satisfies the usual hypotheses
Standard framework for the martingale problem and compensator arguments in Sections 3-4.

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Pith. "Pith review of Derrida-Retaux type models and related scaling limit theorems." pith.science (2026). https://pith.science/paper/EI4VZN6H

@misc{pith2026241112189,
  author       = {Pith},
  title        = {Pith review of: Derrida-Retaux type models and related scaling limit theorems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EI4VZN6H}},
  note         = {Machine review of arXiv:2411.12189}
}

read the original abstract

We give characterizations of the transition semigroup and generator of a continuous-time Derrida--Retaux type process that generalizes the one introduced by Hu, Mallein and Pain (Commun. Math. Phys., 2020). It is shown that the process arises naturally as the scaling limit of the discrete-time max-type recursive models introduced by Hu and Shi (J. Stat. Phys., 2018).

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Forward citations

Cited by 1 Pith paper

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

Asymptotic behavior of the generalized Derrida-Retaux recursive model
math.PR 2024-11 accept novelty 7.0 of 10

For a geometric-offspring generalized Derrida-Retaux model, the marginal parameters have explicit asymptotic expansions in every regime, yielding sharp decay rates for the survival probability and mean.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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and S hi, Z.: Derrida–Retaux conjecture on recursive models

Chen, X.; Dagard, V; Derrida, B.; Hu, Y.; Lifshits, M. and S hi, Z.: Derrida–Retaux conjecture on recursive models . Ann. Probab. 49 (2021), 637–670

work page 2021
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and Shi, Z.: The sustainability probabil ity for the critical Derrida–Retaux model

Chen, X.; Hu, Y. and Shi, Z.: The sustainability probabil ity for the critical Derrida–Retaux model. Probab. Theory Related Fields 182 (2022), 641–684

work page 2022
[5]

and Martin, A.: Stu dy of the iterations of a mapping associated to a spin-glass model

Collet, P.; Eckmann, J.P.; Glaser, V. and Martin, A.: Stu dy of the iterations of a mapping associated to a spin-glass model. Commun. Math. Phys. 94 (1984), 353–370. 27

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[6]

and Retaux, M.: The depinning transition in pr esence of disorder: a toy model

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[9]

and Pain, M.: An exactly solvable conti nuous-time Derrida–Retaux model, Commun

Hu, Y.; Mallein, B. and Pain, M.: An exactly solvable conti nuous-time Derrida–Retaux model, Commun. Math. Phys. 375 (2020), 605–651

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and Shi, Z.: The free energy in the Derrida–Retaux recursive model

Hu, Y. and Shi, Z.: The free energy in the Derrida–Retaux recursive model. J. Stat. Phys. 172 (2018), 718–741

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and Watanabe, S.: Stochastic Diﬀerential Equations and Diﬀusion Processes

Ikeda, N. and Watanabe, S.: Stochastic Diﬀerential Equations and Diﬀusion Processes . 2nd Ed. North-Holland, Amsterdam; Kodansha, Tokyo, 1989

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Aldous, D.: Stopping times and tightness. Ann. Probab. 6 (1978), 335–340

work page 1978

[2] [2]

World scientiﬁc, Sin- gapore, 2004

Chen, M.: From Markov Chains to Non-Equilibrium Particle Systems . World scientiﬁc, Sin- gapore, 2004

work page 2004

[3] [3]

and S hi, Z.: Derrida–Retaux conjecture on recursive models

Chen, X.; Dagard, V; Derrida, B.; Hu, Y.; Lifshits, M. and S hi, Z.: Derrida–Retaux conjecture on recursive models . Ann. Probab. 49 (2021), 637–670

work page 2021

[4] [4]

and Shi, Z.: The sustainability probabil ity for the critical Derrida–Retaux model

Chen, X.; Hu, Y. and Shi, Z.: The sustainability probabil ity for the critical Derrida–Retaux model. Probab. Theory Related Fields 182 (2022), 641–684

work page 2022

[5] [5]

and Martin, A.: Stu dy of the iterations of a mapping associated to a spin-glass model

Collet, P.; Eckmann, J.P.; Glaser, V. and Martin, A.: Stu dy of the iterations of a mapping associated to a spin-glass model. Commun. Math. Phys. 94 (1984), 353–370. 27

work page 1984

[6] [6]

and Retaux, M.: The depinning transition in pr esence of disorder: a toy model

Derrida, B. and Retaux, M.: The depinning transition in pr esence of disorder: a toy model. J. Stat. Phys. 156 (2014), 268–290

work page 2014

[7] [7]

and Meyer, P.A.: Probabilities and Potential

Dellacherie, C. and Meyer, P.A.: Probabilities and Potential . Chapters 5–8. North-Holland, Amsterdam, 1982

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and Kurtz, T.G.: Markov Processes: Characterization and Convergence

Ethier, S.N. and Kurtz, T.G.: Markov Processes: Characterization and Convergence . Wiley, New York, 1986

work page 1986