{"id":"b24c6350-fbcc-45be-a970-7ba212959b69","arxiv_id":"2607.05366","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under bounded weights satisfying Sn/n\toℓ, the depth profile of depth-weighted recursive trees admits an Edgeworth-type scaling limit involving a random analytic function, and depth is a.s. asymptotic to e log n.","lead":"The paper proves a precise scaling limit for the depth profile of depth-weighted random recursive trees when the weights are bounded and obey a law of large numbers. It also supplies a counter-example showing that the depth scaling fails without that condition, answering an open question.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates (1.1) as the weakest (and necessary) assumption and correctly judges the remainder of the argument to be free of critical gaps. The technical novelties—random Zn/hn, control of zeros without self-similarity, and the application of the KMS17 Edgeworth expansion—are executed carefully. Because the full text is available and the estimates close under the stated hypotheses, no adjustment to the ACCEPT verdict is warranted.","tokens_in":25594,"tokens_out":426,"duration_ms":23887,"concrete_test":"Independently re-derive the second-moment bound on log Mn(0) that appears in the display after (3.8) of Lemma 3.7, using only the crude depth bound of Lemma 3.5 and the LLN (1.1); if the resulting series still converges and Fatou still yields M_\rho(0)>0 a.s., the zero-isolation step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 rests on the LLN hypothesis (1.1) together with the boundedness of f, which the paper treats as an explicit hypothesis rather than a hidden gap. The martingale construction (Lemma 2.1), the almost-sure convergence of Mn(z) on E_eta (Proposition 3.4), the isolation of zeros of M_\rho (Lemma 3.7 via L^{2} control of log Mn(0)), the concentration away from the real axis (Lemma 3.8), and the verification of the four KMS17 assumptions (Theorem 3.13) are all carried through under precisely these hypotheses. The counter-example of Proposition 1.4 confirms that (1.1) is necessary, so the hypothesis is load-bearing by design rather than by oversight. No circularity, missing estimate, or unjustified interchange appears in the chain from the martingales to the Edgeworth expansion.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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 \to 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 \to 1 and d(Tn)/log n \to 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.","tokens_in":25848,"tokens_out":863,"duration_ms":6622,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Theorem 1.1"},{"comment":"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.","section":"Lemma 3.7"},{"comment":"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.","section":"Introduction"},{"comment":"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).","section":"Throughout"}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for publication essentially as is. The only non-trivial open point is the possible presence of zeros of W∞, which the authors correctly flag as open; it does not affect any of the stated theorems. Fit for a probability journal is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives the first scaling limit of the full depth profile for depth-weighted recursive trees when f is bounded above and below and Sn/n converges. That immediately yields d(Tn)/log n \to e a.s., generalising the convergent/periodic cases of Lichev–Linker–Lodewijks–Mitsche, and the two-value counter-example shows the LLN is necessary, answering their open Question 1.5 negatively.\n\nWhat is new is the appearance of the Sk term in the local limit (so the profile can jump by a factor f(k) from one generation to the next) and the fact that the argument works with random Zn and hn. The author adapts Sénizergues’ martingale + KMS17 Edgeworth route, but has to prove martingale convergence on half-planes, isolate zeros of the limiting analytic function without self-similarity (by L^{2} control of log Mn(0)), and get concentration off the real axis. All four KMS hypotheses are checked carefully; the proofs look complete and the citation pattern is appropriate.\n\nSoft spots are minor and acknowledged. The LLN is load-bearing by design (the counter-example confirms it), and the author leaves open whether W∞ has zeros on (−\neq,1). The Edgeworth expansion is used as a black box after the hypotheses are verified, which is fine. No circularity or missing estimates jumped out.\n\nThis is for people who work on recursive trees, branching random walks, or profile asymptotics. A serious referee will want to check the zero-isolation argument and the random-hn estimates, but the paper is already in good shape. I would send it out and I would cite the profile limit and the counter-example.","headline":"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.","tokens_in":26425,"tokens_out":467,"would_cite":true,"duration_ms":5061,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60C05","05C05","60F05"],"pacs":[],"model":"grok-4.5","headline":"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.","keywords":["depth-weighted recursive trees","depth profile","scaling limit","martingales","Edgeworth expansion","random recursive trees","height of trees"],"falsifier":"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.","tokens_in":26508,"feed_emoji":"🌳","tokens_out":987,"duration_ms":14458,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bounded weights force tree depth to e log n","feed_subtitle":"A law of large numbers on log-weights is necessary; without it the depth ratio can oscillate.","key_machinery":"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.","core_discovery":"Under the sole assumptions that c ≤ f ≤ 1/c and Sn/n \to 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 \to 1 and d(Tn)/log n \to e almost surely.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Bounded weights pin recursive tree depth to e log n","Depth profiles scale as e log n under weight LLN","Bounded weights force almost-sure depth limit e log n","Tree depth ratio reaches e when Sn/n vanishes","Scaling limit of depth fails without weight LLN"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The cumulative log-weights must grow linearly (Sn/n converges); without that average the depth ratio can oscillate between two different constants.","fun_headline_variants_meta":{"raw":{"variants":["Bounded weights pin recursive tree depth to e log n","Depth profiles scale as e log n under weight LLN","Bounded weights force almost-sure depth limit e log n","Tree depth ratio reaches e when Sn/n vanishes","Scaling limit of depth fails without weight LLN"]},"model":"grok-4.5","effort":"low","cost_usd":0.004642,"raw_usage":{"total_tokens":1307,"prompt_tokens":704,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":46420000,"prompt_tokens_details":{"text_tokens":704,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":524,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":704,"tokens_out":79,"duration_ms":4226,"temperature":1.0,"reasoning_tokens":524,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T07:13:45.870498+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":2}