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Depth profile of depth-weighted trees with bounded weights

T0 review · 0 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read When depth weights stay bounded and obey a law of large numbers, the profile of a recursive tree converges to a Gaussian times an exponential weight factor and the depth is almost surely e log n.

desk verdict Solid profile scaling limit for depth-weighted trees under LLN+bounded weights, plus a clean negative answer to LLLM26 Question 1.5; the technical work is careful and self-contained. read the letter →

arxiv 2607.05366 v1 pith:MBWBJTOF submitted 2026-07-06 math.PR

Emmanuel Kammerer This is my paper
classification math.PR MSC 60C0505C0560F05
keywords depth-weightedrecursivetreesdepthprofilescalinglimitmartingalesEdgeworthexpansionrandomheightof
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

Depth-weighted recursive trees attach each new vertex with probability proportional to a weight that depends only on the depth of the attachment point. When those weights are bounded away from zero and infinity and their logarithms satisfy a law of large numbers, the paper proves a precise scaling limit for the whole depth profile. The number of vertices at depth k is asymptotically a random multiple of a Gaussian density centred at a random height hn that itself is asymptotic to log n, multiplied by the cumulative product of the weights. As an immediate corollary the height of the tree is almost surely asymptotic to e log n, extending earlier results that required the weights to be constant, periodic or slowly varying. The same argument also produces a simple two-valued counter-example showing that the e-log-n law can fail as soon as the law-of-large-numbers assumption is dropped.

What carries the argument

Complex martingales Mn(z) obtained by normalising the weighted Laplace transform of the profile by the product Cn(z) = ∏(1 + e^z/Zk); their almost-sure analytic convergence on a half-plane, combined with the general Edgeworth expansion of Kabluchko–Marynych–Sulzbach, yields the local limit for the profile.

What would settle it

Take any bounded weight sequence for which Sn/n fails to converge (for instance the explicit two-valued function of Proposition 1.4) and check whether lim d(Tn)/log n exists; the paper already exhibits liminf ≤ 5/2 < e ≤ limsup almost surely.

Watch

Extended reading notes

Core claim

Under the sole assumptions that c ≤ f ≤ 1/c and Sn/n o 0, the depth profile admits the almost-sure local limit Ln(k) = W∞(0) e^{hn}/√(2π hn) exp(Sk - ½((k-hn)/√hn)^{2}) + e^{Sk} o(e^{hn}/√hn) uniformly in k, while hn/log n o 1 and d(Tn)/log n o e almost surely.

Load-bearing premise

The cumulative log-weights must grow linearly (Sn/n converges); without that average the depth ratio can oscillate between two different constants.

Editorial extensions

If this is right

  • The classic e-log-n height law holds for every bounded weight sequence whose logs obey a law of large numbers, including i.i.d. random weights.
  • The profile can jump by a factor f(k) from one generation to the next even inside the bulk of the tree.
  • A typical vertex still sits at depth ~ log n, so the diameter between two typical vertices is expected to be ~ 2 log n.
  • The same martingale-plus-Edgeworth route applies verbatim to slowly-growing or polynomial weights once the total weight sum is understood.

Reading between the lines

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

  • The zeros of the limiting analytic function W∞ may still be empty almost surely; proving that would remove the exceptional set that currently appears in the local-limit statement.
  • The second-order correction to the height is probably of order max |Sk| up to depth e log n, which for i.i.d. weights would be of order √log log n.
  • The same Laplace-transform martingales should control the profile for preferential-attachment trees once they are rewritten as weighted recursive trees.
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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

0 major / 4 minor

Summary. The paper studies depth-weighted random recursive trees with weights f bounded above and below by a positive constant. Under the LLN hypothesis Sn/n o ℓ (taken w.l.o.g. as 0), it proves an almost-sure local limit for the depth profile Ln(k) involving a random analytic factor W∞(0) and the random sum hn of inverse total weights (Theorem 1.1). As corollaries one obtains hn/log n o 1 and d(Tn)/log n o e a.s., together with the typical-depth statement of Theorem 1.3. The argument constructs the weighted Laplace transforms Ln(z), the associated martingales Mn(z) = Ln(z)/Cn(z), establishes their a.s. and L1 convergence on suitable half-planes (Prop. 3.4), isolates the zeros of the limit via L2 control of log Mn(0) (Lemma 3.7), obtains concentration off the real axis (Lemma 3.8), verifies the four hypotheses of the Kabluchko–Marynych–Sulzbach Edgeworth expansion, and recovers the profile asymptotics. Proposition 1.4 supplies an explicit two-value counter-example showing that the depth scaling fails without the LLN assumption, answering a question of Lichev et al.

Significance. The work closes a natural open question on the depth of depth-weighted trees under the mild LLN condition that covers all previously treated convergent, periodic and i.i.d. cases, while simultaneously giving a sharp profile limit that exhibits the novel Sk correction. The negative answer furnished by the two-value counter-example is clean and definitive. Methodologically the paper demonstrates that the general Edgeworth machinery of KMS17 can be applied once suitable martingales are controlled, and it carefully overcomes the new difficulties arising from the randomness of Zn and hn. The results are self-contained, the hypotheses are sharp, and the proofs are complete; the paper therefore constitutes a solid and useful contribution to the asymptotic theory of recursive trees.

minor comments (4)
  1. [Theorem 1.1] In the statement of Theorem 1.1 the o-term is claimed to be uniform in k∈N; a short parenthetical remark that the uniformity follows from the KMS17 remainder estimate would help the reader.
  2. [Lemma 3.7] Lemma 3.7 controls log Mn(0) only on the events Bk; while the argument is correct, a one-sentence reminder that igcup Bk has full probability (by Lemma 3.5 and Corollary 3.6) would make the passage to M∞(0)>0 a.s. more transparent.
  3. [Introduction] The open problem left at the end of the introduction (whether W∞ has zeros on (-∞,1)) could be restated more prominently, perhaps as a separate Question, so that it is not overlooked.
  4. [Throughout] A few minor typos: “T echniques” and “F urther directions” in the introduction; “we have1 +ez/Zn” missing spaces (p.5); “for allnlarge enough” (several places).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: martingales and Edgeworth application are self-contained under explicit hypotheses

full rationale

The derivation constructs the weighted Laplace transform Ln(z) and the normalizing product Cn(z) directly from the recursive attachment rule (Lemma 2.1), obtains Mn(z) as a martingale, proves its a.s. and L1 convergence on suitable half-planes by moment estimates that use only the boundedness of f and the LLN hypothesis (1.1) (Proposition 3.4, Lemmas 3.1–3.2), controls the zeros of the limit by an L2 bound on log Mn(0) (Lemma 3.7), establishes concentration off the real axis (Lemma 3.8), and verifies the four hypotheses of the external Edgeworth theorem of Kabluchko–Marynych–Sulzbach (Theorem 3.13). All objects (hn, W∞, Z) are defined from the tree process itself; no parameter is fitted to data and then re-used as a prediction, and the sole external black-box result is an independent analytic theorem applied after its assumptions have been checked. The counter-example of Proposition 1.4 further shows that (1.1) is necessary rather than smuggled. The chain therefore contains no self-definitional, fitted-input, or load-bearing self-citation circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper is a pure-probability derivation. The only external inputs are standard martingale theory, the cited Edgeworth expansion of KMS17, and the modelling assumptions (bounded weights and the LLN condition) that define the regime. No free parameters are fitted; the random analytic function W∞ is obtained as an a.s. limit rather than postulated.

assumptions (3)
  • domain assumption Weights satisfy c≤f(n)≤1/c for some deterministic c>0 and all n.
    Stated at the outset of Section 1 and used for all moment comparisons (e.g., (1.3), Lemma 2.2).
  • domain assumption The partial sums Sn=∑_{k=0}^{n-1} log f(k) obey Sn/n oℓ∈R (taken w.l.o.g. as 0).
    Hypothesis (1.1); required for the main theorems and for the control of e^{-Sd} in Lemmas 3.1 and 3.5.
  • standard math The general Edgeworth expansion of Kabluchko-Marynych-Sulzbach (Theorem 2.1 of KMS17) applies once A1–A4 are verified.
    Invoked in Subsection 3.4 after the four hypotheses are checked; treated as a black-box external theorem.
invented entities (1)
  • Random analytic function W∞(z)=e^{c(z)}M∞(z) on the half-plane Re z<1
    purpose: Appears as the almost-sure limit of the normalized Laplace transform and multiplies the Gaussian profile in the local limit.
    Constructed as the uniform limit of the martingales Mn(z); existence and analyticity are proved in Proposition 3.4, positivity at 0 in Lemma 3.7. No independent physical or combinatorial interpretation is claimed outside the tree process.

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Pith. "Pith review of Depth profile of depth-weighted trees with bounded weights." pith.science (2026). https://pith.science/paper/MBWBJTOF

@misc{pith2026260705366,
  author       = {Pith},
  title        = {Pith review of: Depth profile of depth-weighted trees with bounded weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBWBJTOF}},
  note         = {Machine review of arXiv:2607.05366}
}
read the original abstract

We study the depth-weighted random recursive trees introduced by Leckey, Mitsche and Wormald in the case where the weights are bounded from above and from below. We establish the scaling limit of the depth profile of these trees when the weights satisfy a law of large numbers. In particular, we obtain the scaling limit of the depth of these trees, generalising some results of Lichev, Linker, Lodewijks and Mitsche. We also answer negatively a question left open by the same authors, showing that the scaling limit of the depth does not hold in general. Our main tools are appropriately defined martingales combined with the general Edgeworth expansion for the profiles introduced by Kabluchko, Marynych and Sulzbach.

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.