Asymptotic behavior of the generalized Derrida-Retaux recursive model

Run Zhang; Zenghu Li

REVIEW 2 major objections 5 minor 1 cited by

Asymptotic behavior of the generalized Derrida-Retaux recursive model

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives sharp asymptotic expansions for the parameters of the generalized Derrida–Retaux model with geometric offspring, including a critical regime with a log n/n^2 correction, and obtains the sustainability probability to…

desk verdict Solid, self-contained contribution to the Derrida–Retaux program, but the advertised critical log correction has a sign error that makes Theorem 1.3(3) and the critical corollaries false as stated. read the letter →

arxiv 2411.13068 v3 pith:5HUERBM5 submitted 2024-11-20 math.PR

Zenghu Li , Run Zhang This is my paper · ORCID

classification math.PR MSC 60J0582B27

keywords GeneralizedDerrida–Retauxmodelgeometricoffspringdistributiongeometric-typesustainabilityprobabilityfirstmomentgeneratingfunctionasymptoticbehavior

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 studies the generalized Derrida–Retaux recursion $Y_{n+1}\overset{d}{=}(Y_{n,1}+\cdots+Y_{n,\eta}-1)_+$ when the offspring number $\eta$ has the geometric law (1.6) with mean $m>1$. It proves that if the initial state has a geometric-type distribution, every later marginal has the same two-parameter form, and it derives expansions for the two parameters $r_n,p_n$ in the supercritical, subcritical, and critical regimes. The critical case is the sharpest: $\mathbb{P}(Y_n\ge 1)$ decays as $2/((m-1)^2n^2)+8(m+1)\log n/(3(m-1)^3n^3)+o(\log n/n^3)$. These formulas correct the prefactors conjectured for deterministic offspring number $m$, because the random offspring count slows mass production.

What carries the argument

The central object is the two-parameter family of geometric-type laws $G(r,p)$, with atom $p$ at the origin and a geometric tail with ratio $1-r$ on the positive integers. The load-bearing identity is the recurrence (1.7) and its consequence (2.8), $$\frac1{r_{n+2}}-\frac1{r_{n+1}}=m(1-r_{n+1})\left(\frac1{r_{n+1}}-\frac1{r_n}\right),$$ which lets the authors convert local difference estimates into global asymptotic expansions by the Stolz–Cesàro theorem. In the critical case the proof works with the displacement $v_n=r_n-(1-m^{-1})$ and the identity (3.18), which isolates the excess of $1/v_{n+1}-1/v_n$ over $m/2$; this is what produces the logarithmic correction.

What would settle it

Iterate the recurrence (1.7) numerically for $m=2$ from any initial pair whose limits are $r_*=1/2$ and $p_*=1$, and compute $n^3(v_n-v_{n+1}-v_n v_{n+1})$ with $v_n=r_n-1/2$; Theorem 1.3 requires this to converge to $2$, so any other long-run value would refute the expansion.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: for geometric offspring, the two-parameter geometric-type family $G(r,p)$ is preserved under the recursion, and the parameters converge to one of three regimes. In the critical regime $r_*=1-m^{-1}$, $p_*=1$, the parameters obey $$r_n = 1-\frac1m+\frac{2}{mn}-\frac{4(m+1)\log n}{3m(m-1)$n^{2}$}+o\!\left(\frac{\log n}{$n^{2}$}\right),\qquad p_n = 1-\frac{2}{(m-1)^$2n^{2}$}-\frac{8(m+1)\log n}{3(m-1)^$3n^{3}$}+o\!\left(\frac{\log n}{$n^{3}$}\right).$$ From these expansions the paper derives the sharp sustainability probability, the conditional geometric limit law, the first moment, and the probability generating function. The critical conditional law matches the earlier deterministic-offspring conjecture, but the decay prefactors differ because of randomness in the number of offspring.

Load-bearing premise

Everything rests on the offspring law being exactly geometric, $\mathbb{P}(\eta=n)=(1/m)(1-1/m)^{n-1}$; this is what keeps every marginal inside the two-parameter geometric-type family, and without it the recurrence (1.7) and all expansions that follow would not hold.

Editorial extensions

If this is right

In the critical regime the sustainability probability obeys $\mathbb{P}(Y_n\ge1)=2/((m-1)^2n^2)+8(m+1)\log n/(3(m-1)^3n^3)+o(\log n/n^3)$.
The conditional law $\mathbb{P}(Y_n=k\mid Y_n\ge1)$ converges to $(1-1/m)(1/m)^{k-1}$ in the critical case, matching the deterministic-offspring conjecture (C2).
The first moment in the critical case is $\mathbb{E}(Y_n)=2m/((m-1)^3n^2)+8m(m+1)\log n/(3(m-1)^4n^3)+o(\log n/n^3)$.
The critical generating function at $s=m$ behaves like $\mathbb{E}(m^{Y_n})=1+1/((m-1)n)+2(m+1)\log n/((m-1)^2n^2)+o(\log n/n^2)$.
The prefactors differ systematically from the fixed-offspring conjectures (C1), (C3), (C4a), (C4b), and the paper attributes the slowdown to randomness in the offspring number.

Reading between the lines

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

The same two-parameter closure and Stolz–Cesàro summation should transfer to the exponential-type marginals mentioned in Remark 1.12, yielding analogous critical $(\log n)/n^2$ corrections in the continuous-time analogue.
The geometric-law prefactors likely encode the variance of the offspring distribution; comparing the coefficients with the deterministic $m$ case suggests a testable dependence on the second moment of $\eta$.
The remainder constant in (3.19), namely $4(m+1)/(3m(m-1))$, is a sharp numerical diagnostic: for $m=2$ it equals $2$, so direct iteration of the recurrence provides an unambiguous check of the whole asymptotic program.

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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

Here's my take on Li–Zhang (arXiv:2411.13068). The paper is real work: it shows the geometric-type family is closed under the generalized DR recursion with geometric offspring, writes the recursion as a two-parameter map (1.7), and then extracts sharp asymptotics for the supercritical, subcritical, and critical regimes. The super- and subcritical expansions look right to me, the proofs are detailed, and the authors are upfront about the limits of the geometric assumption and about the concurrent work by Alsmeyer–Hu–Mallein. I agree with your positive read on the method.

But there is a sign error in the critical case that you missed. The theorem states p_n = 1 - 2/((m-1)^2 n^2) - 8(m+1)/(3(m-1)^3) log n/n^3 + o(log n/n^3). Use their own identity (2.9): 1 - p_n = (r_n - r_{n+1})/((m-1) r_{n+1}). With v_n = 2/(mn) - A log n/n^2, A = 4(m+1)/(3m(m-1)), you get v_n - v_{n+1} = 2/(m n^2) - 2A log n/n^3 + o. Plugging into (2.9) gives 1-p_n = 2/((m-1)^2 n^2) - 8(m+1)/(3(m-1)^3) log n/n^3 + o. Hence p_n = 1 - 2/((m-1)^2 n^2) + 8(m+1)/(3(m-1)^3) log n/n^3 + o. The proof's own display for 1-p_n in the critical section carries the minus sign in the log term, so the proof line is right and the theorem statement is wrong. Consequently Corollary 1.5(3), Corollary 1.8(3), and Corollary 1.9(3) have the wrong sign on the log n/n^3 term as well.

This is a load-bearing error because the log correction is exactly the advertised sharp output, but it is mechanical and fixable. The leading-order constants and the structure of the argument are unaffected. If the authors flip the sign in the statement and corollaries and double-check the affected derivations, the paper will be fine. As it stands, I would not cite the critical-case formulas.

Overall: serious paper, honest about its scope, but it needs a correction before the critical results are used. I'd send it back for minor revision rather than desk-reject or accept as-is.

Referee Report

2 major / 5 minor

Summary. The paper studies the generalized Derrida–Retaux process (1.3) under a geometric offspring distribution (1.6). It proves that the class of geometric-type marginal distributions is preserved, reducing the model to the two-parameter recursion (1.7), and it characterizes the three regimes through the limits (r*, p*). The main results are asymptotic expansions of r_n and p_n in the supercritical, subcritical, and critical cases (Theorem 1.3), with consequences for the sustainability probability, first moment, and probability generating function (Corollaries 1.5, 1.8, 1.9) and for the conditional limit laws (Corollary 1.6). The critical-case analysis contains nontrivial logarithmic corrections, and the proofs are based on generating-function recursions, the preservation theorem, and Stolz–Cesàro limit arguments.

Significance. If the stated results were correct, the paper would give the sharp critical scaling for the geometric-offspring version of the model, including the constants in front of the log n/n^2 and log n/n^3 corrections, and would refine the conjectures (C1)–(C4b) for this solvable family. The proof strategy is largely self-contained, does not use fitted parameters, and the preservation theorem for geometric-type distributions is elegant. However, the advertised asymptotic constants contain sign and algebraic errors in two central places, so the paper cannot be accepted without correction.

major comments (2)

[Theorem 1.3(3), Corollaries 1.5(3), 1.8(3), 1.9(3)] Section 3, Theorem 1.3(3): the sign of the log n/n^3 term in the critical p_n expansion is wrong. From (3.21), v_n = r_n - (1 - 1/m) = 2/(mn) - A log n/n^2 with A = 4(m+1)/(3m(m-1)); hence v_n - v_{n+1} = 2/(m n^2) - 2A log n/n^3 + o(log n/n^3), as in (3.23). Identity (2.9) then gives 1 - p_n = (v_n - v_{n+1})/((m-1)(1 - 1/m + v_{n+1})) = 2/((m-1)^2 n^2) - 8(m+1)/(3(m-1)^3) log n/n^3 + o(log n/n^3), so p_n = 1 - 2/((m-1)^2 n^2) + 8(m+1)/(3(m-1)^3) log n/n^3 + o(log n/n^3). The theorem states the opposite sign, while the proof's own displayed line after (3.24) has the correct minus sign in 1 - p_n. Consequently Corollary 1.5(3) should have the negative log term, Corollary 1.8(3) should have the negative log coefficient, and both displays in Corollary 1.9(3) need corrected signs; in particular (1.15) should read E(m^{Y_n}) = 1 + 1/((m-1)n) - 2(m+1)/(3(m-1)^2) log n/n^2 + o(log n/n^2).
[Theorem 1.3(2), Corollaries 1.5(2), 1.8(2), 1.9(2)] Section 3, Theorem 1.3(2): the second-order subcritical coefficients are algebraically wrong. Using (3.8), d_n = r_n - r_{n+1} = K_n gamma_*^n r_n r_{n+1}, with K_n = K + mK^2 r_*^2/(1 - gamma_*)^2 gamma_*^n + o(gamma_*^n) and r_n = r_* + K r_*^2/(1 - gamma_*) gamma_*^n + o(gamma_*^n), gives d_n = K r_*^2 gamma_*^n + K^2 r_*^3 (1 - gamma_*^2 + m r_*)/(1 - gamma_*)^2 gamma_*^{2n} + o(gamma_*^{2n}). Summing yields r_n = r_* + K r_*^2 gamma_*^n/(1 - gamma_*) + K^2 r_*^3 (1 - gamma_*^2 + m r_*)/[(1 - gamma_*)^2 (1 - gamma_*^2)] gamma_*^{2n} + o(gamma_*^{2n}), not the displayed (1 + m r_*)/[(1 - gamma_*^2)(1 - gamma_*)^2]. Equivalently, from (2.9), 1 - p_n = K r_* gamma_*^n/(m - 1) + K^2 r_*^2 (1 - gamma_* + m r_*)/[(m - 1)(1 - gamma_*)^2] gamma_*^{2n} + o(gamma_*^{2n}), so p_n should have minus this second term. The same correction propagates to Corollaries 1.5(2), 1.8(2), and 1.9(2); for example, the gamma_*^{2n} coefficient in Corollary 1.8(2) should be mK^2 r_*^2/[(m - 1)(1 - gamma_*)^2], not mK^2 r_*/[(m - 1)(1 - gamma_*)^2].

minor comments (5)

[Lemma 3.2] The opening sentence contains a typo: 'The the approach' should read 'The approach'.
[Proof of Theorem 1.3(3)] The displayed identity after (3.23) writes m/2 v_n v_{n+1} = m/2 v_n^2 + m/2 v_n(v_n - v_{n+1}); the sign before the second term should be minus. This does not affect the order shown, but the line should be corrected.
[Lemma 3.1] The statement 'v_n strictly decreases to zero' should be qualified: in the critical case r_n decreases to r_*, so v_n = r_n - r_* decreases to zero from above; if the initial value put v_n below zero, the wording needs an index shift.
[Remark 1.11] The comparison of the critical constants with the conjectures (C1), (C3), (C4a), and (C4b) should be revisited after the sign corrections, since the sign of the logarithmic correction changes the qualitative comparison.
[References and formatting] Reference [15] and the proof of Corollary 1.8 contain missing spaces ('Derrida–Retauxtypemodels', 'Inthesubcriticalcase'), and the proof of Corollary 1.6(1) would benefit from a brief justification of the interchange of limits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic expansions are solved from the exact recursion, not assumed or fitted.

full rationale

The paper's derivation chain is self-contained. Theorem 1.1 proves, rather than assumes, the closure of geometric-type laws under the recursion, using the generating function identity (2.1) and the geometric offspring law (1.6). Proposition 2.2 derives the key exact difference equation (2.8), and Theorem 1.3 solves this equation via the Stolz–Cesàro theorem and Taylor expansions. The constants F∞ and K are defined by convergent infinite products over the already-converging sequences (1.9) and (1.10), so no fitted or externally imposed parameter is renamed as a prediction. The corollaries are exact rearrangements: P(Yn ≥ 1) = 1 − pn, E(Yn) = (1 − pn)/rn, and the generating function formula (2.1), so they are consequences of Theorem 1.3 rather than inputs to it. The only self-citation, [15] in the introduction, is contextual and not load-bearing for any theorem. No uniqueness theorem from the authors' prior work is invoked to force a choice, and no ansatz is imported by citation. A skeptical reviewer has noted a possible sign error in the log n/n^3 coefficient of pn in Theorem 1.3(3); if real, that is an arithmetic/correctness flaw, but it is not circularity because the displayed expansion is still derived from (2.9), (3.21), and (3.23), not assumed. The analysis therefore earns a score of 0.

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

The central claims rest only on the model assumptions (geometric offspring, geometric-type initial law, independence) and standard analytic tools. The asymptotic constants F∞, Q, K and γ_* are defined as limits/infinite products of the sequence itself, not fitted or chosen ad hoc. No free parameters or new entities are introduced.

assumptions (5)

standard math Stolz-Cesàro theorem is applicable to the difference sequences derived from (1.7).
Used throughout Section 3 (e.g., proofs of Theorem 1.3(1)-(3), Lemma 3.1) to convert difference equations into limits of ratios.
standard math Taylor expansions of exp, log, and 1/(1±x) hold for the small remainder terms.
Used repeatedly in the asymptotic expansions, e.g., expanding exp(-n/(F∞ m^{n-1})) and the log product in the supercritical case (Section 3).
standard math Borel-Cantelli lemma for almost sure convergence of Y_n.
Used in the proof of Corollary 1.6 to show Y_n → ∞ (supercritical) or Y_n → 0 (subcritical/critical) from summability of tail probabilities.
domain assumption The offspring number η has a geometric distribution with mean m: P(η=n) = (1/m)(1-1/m)^{n-1}, n ≥ 1.
Equation (1.6) is the defining assumption of the model and is essential for the preservation of geometric-type marginals proved in Theorem 1.1.
domain assumption The initial distribution is µ0 = G(r0,p0) with (r0,p0) ∈ (0,1)^2, and {Y_{n,k}} are independent copies of Y_n independent of η.
These independence and initial-condition assumptions are stated in the model (1.3) and Theorem 1.1; they set the stage for the generating function computation (2.1).

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Pith. "Pith review of Asymptotic behavior of the generalized Derrida-Retaux recursive model." pith.science (2026). https://pith.science/paper/5HUERBM5

@misc{pith2026241113068,
  author       = {Pith},
  title        = {Pith review of: Asymptotic behavior of the generalized Derrida-Retaux recursive model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HUERBM5}},
  note         = {Machine review of arXiv:2411.13068}
}

read the original abstract

We study the max-type recursive model introduced by Hu and Shi (J. Stat. Phys., 2018), which generalizes the model of Derrida and Retaux (J. Stat. Phys., 2014). The class of geometric-type marginal distributions is preserved by the model with a geometric offspring distribution. We give some long-time asymptotic expansions of the parameters of the marginal distribution. From the expansions, we derive the asymptotics of the sustainability probability, marginal distribution, first moment and probability generating function.

Discussion (0). Continue with ORCID to comment.

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

Derrida-Retaux type models and related scaling limit theorems
math.PR 2024-11 conditional novelty 6.0 of 10

Rescaled generalized Derrida-Retaux dynamics converge in Skorokhod space to a continuous-time process whose semigroup, generator, and martingale problem are characterized.

Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages · cited by 1 Pith paper

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Pith tools

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

[1] [1]

and Bandyopadhyay, A.: A survey of max-type recursive distributional equations

Aldous, D.J. and Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15, 1047–1110 (2005)

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

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

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