{"id":"dcabcd00-a922-4614-be97-b1f648e5285e","arxiv_id":"2412.08236","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Trap models on convergent sequences of resistance networks converge and exhibit aging; with local-structure convergence they also exhibit sub-aging, covering Sierpinski gaskets, critical Galton-Watson trees, and critical Erdos-Renyi graphs.","lead":"This mathematics paper proves that Bouchaud trap models, random walks with heavy-tailed waiting times, show aging on convergent sequences of resistance networks, including Sierpinski gaskets and critical random graphs. It also introduces a topological framework that turns network convergence into convergence of the aging and sub-aging functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof of Theorems 1.6/1.9 is internally consistent. The main caveat is the dependence on unpublished preprints [37]-[39] and the sketched verification in §7.2, not a demonstrated flaw.","rationale":"The reader's CONDITIONAL verdict is reasonable. I focused on Assumption 1.5(ii) because it is indeed the entry point of [21, Theorem 1.2] and is used in Lemma 6.10 to control truncated processes, but I could not establish that it is an actual weak point of the argument: it is a stated hypothesis, and the applications provide plausible verification (Sierpinski directly, GW via Corollary 7.4, ER via Theorem 7.6). The sketch in §7.2 is the least complete application, but it is not the central theorem. The largest unverified component is the author's own unpublished framework [37,38,39], whose correctness the paper assumes. No internal inconsistency or counterexample surfaced in my reading; hence the verdict should remain unchanged.","tokens_in":70880,"tokens_out":36432,"duration_ms":435973,"concrete_test":"Independently verify the two load-bearing external inputs: (1) re-derive Theorem 2.21 (Polishness of Mdis(S)) and Theorem 3.17 (Polishness of Gromov-Hausdorff-type topologies) from Sections 2-3 without citing [38]; (2) in the random-conductance model §7.2, prove the claimed convergence (V_Gn,2^{-n}R_Gn,ρ,2^{-n}dot-mu#_Gn) → (R,d_R,0,Leb⊗P(ζ_0+ζ_1∈·)) directly from [39, Theorem A.2], including tightness in the conductance coordinate. If both pass, the main theorems stand as proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Reading the proof in good faith, the central claim rests on four inputs: the Polish metrization of the vague-and-point-process topology (Section 2, based on [38]), the trap coupling Lemma 6.3, the deterministic convergence Theorem 5.7, and the process-convergence theorem [21, Theorem 1.2] under Assumption 1.5(ii). I checked the internal chain: Lemma 6.3 correctly upgrades vague convergence to point-process convergence via Theorem 2.26 and Corollary 2.27; Proposition 4.13 supplies the transition-density precompactness used in Lemmas 5.3-5.4; Lemma 6.10 is a standard truncation argument and its double-limit can be made valid by choosing T=λ^{1/2} before letting δ↓0. I could not find an equation with a hidden assumption or a sequence satisfying all assumptions for which the conclusion fails. The genuinely load-bearing unverified inputs are external correctness of [37,38,39] and the brief assertion in Section 7.2 that the marked-measure convergence holds. These justify a conditional rather than unconditional verdict, but they do not amount to an internal objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves convergence and aging results for symmetric Bouchaud trap models on sequences of electrical networks that converge in the local Gromov-Hausdorff-vague topology and satisfy a uniform non-explosion condition. The main theorems (1.6 and 1.9, with random counterparts 1.13 and 1.16) identify the scaling limits of the traps, the time-scaled BTM processes, and the corresponding aging and sub-aging functions. The proofs are organized around two new technical ingredients: a Polish metrization of the vague-and-point-process topology on discrete measures (Section 2) and a precompactness result for transition densities of processes on resistance metric spaces (Proposition 4.13). The results are then specialized to the Sierpinski gasket, the one-dimensional random conductance model, critical Galton-Watson trees, and the critical Erdos-Renyi random graph.","tokens_in":71113,"tokens_out":23037,"duration_ms":240160,"significance":"If the results are correct, they significantly extend the scope of aging theorems for trap models, replacing the uniform volume-doubling condition of Croydon-Hambly-Kumagai with the weaker non-explosion condition and adding sub-aging limits. The limit objects are explicit: the traps converge to a Poissonian random measure on the limiting resistance space, and the aging functions are described by the associated diffusion. The paper contains original, reusable machinery, notably the Polish metrization and tightness criteria for the vague-and-point-process topology (Theorems 2.21, 2.22, 2.26) and the deterministic convergence theorems in Section 5. The applications to critical Galton-Watson trees and critical random graphs give new aging and sub-aging limits for natural low-dimensional random graphs. No fitting parameters appear, and the limits are derived from the model rather than imposed.","major_comments":[{"comment":"The central proof chain depends at several load-bearing points on the author's unpublished preprints [37], [38], and [39]. Examples include: the vague-metric results underpinning Section 2 (Theorem 2.2 and Proposition 2.3 from [38]); the complete functor metrization framework of Section 3, including Theorems 3.10, 3.13, 3.17, and 3.21 from [38]; the regularity and recurrence facts for resistance forms used after Assumption 1.5 and in Section 4, including [39, Corollary 3.22] and [39, Theorem 5.1]; and the measurability claim in the proof of Lemma 6.10, which cites [37, Lemma 6.3]. Because these are not yet refereed and are not reproduced in the manuscript, the validity of Theorems 1.6, 1.9, 1.13, and 1.16 cannot be independently assessed from the present submission. Please either include statements (or proofs) of these imported results in an appendix, or otherwise make the paper self-contained for every claim that is used in the main convergence chain.","section":"Sections 1-6, general"},{"comment":"The random conductance model application is not fully supported. After citing [39, Theorem A.2] for the resistance-metric convergence, the paper asserts, without proof, that the marked-measure convergence (V_{G_n}, 2^{-n}R_{G_n}, \\rho_{G_n}, 2^{-n}\\dot{\\mu}^#_{G_n}) \\to (\\mathbb{R}, d_{\\mathbb{R}}, 0, \\mathrm{Leb} \\otimes P(\\zeta_0+\\zeta_1\\in\\cdot)) holds. This is precisely Assumption 1.8 and is load-bearing for the advertised sub-aging result. Please supply a proof or a precise reference. In addition, the last display in this subsection cites Theorem 1.9 where Theorem 1.16 is meant.","section":"Section 7.2"},{"comment":"Equation (5.1) is stated as a consequence of weak convergence, but the estimate is uniform in n and in the limit r\\to\\infty. Please expand the proof: either explain how the uniformity follows from the Skorohod convergence together with recurrence of the limit (control of the finitely many small n), or state the non-explosion condition as an explicit hypothesis in Assumption 5.1. As written, this lemma is the entry point for the equicontinuity arguments in Lemmas 5.3 and 5.4, so the missing justification is load-bearing for Theorem 5.7.","section":"Section 5.1.1, Lemma 5.2"}],"minor_comments":[{"comment":"The sentence that begins \"To apply the sub-aging result (Theorem 1.13)\" should refer to Theorem 1.16, not Theorem 1.13.","section":"Section 7.3"},{"comment":"The proof skips the normalization bookkeeping: it first obtains convergence with m_n^{-1}\\dot{\\mu}^#_{\\tilde T} to \\sigma^{-1}\\mu_{2\\tilde e(\\sigma)}\\otimes\\tilde p, and then converts this to the n^{-2/3} scaling in Theorem 7.6. Please spell out the factor \\sigma when applying Theorem A.13, so the reader can verify that the limit measure is \\mu_{M^{(Z_1)}}\\otimes\\tilde p and not a scalar multiple of it.","section":"Section 7.4, proof of Theorem 7.6"},{"comment":"In the proof of Lemma 6.5, the notation in the condition \"\\mu(\\partial(A\\times(u_1,\\infty)))=0\" should be \\dot{\\mu}, since the measure at that point is the marked measure on F\\times\\mathbb{R}_{\\ge0}.","section":"Lemma 6.5"},{"comment":"The text says that Theorem 7.6 shows Assumption 1.15 holds, but Assumption 1.15 also includes the non-explosion condition (Assumption 1.12(ii)). Please add a sentence explaining why the compactness of the limiting fused space implies the required non-explosion condition for the sequence of largest components.","section":"Section 7.4, before Theorem 7.6"},{"comment":"The spaces in which the convergences in Theorems 1.6 and 1.9 take place are written out in full and are very hard to parse. Introducing short names for the functors and associated metric spaces in Section 1, or collecting them in a table after Section 3, would materially improve readability.","section":"Section 1, Theorems 1.6 and 1.9"}],"recommendation":"major_revision","confidential_remarks":"The main uncertainty is external: the proof depends on three unpublished preprints by the same author ([37], [38], [39]). If the editor has verified those preprints or has referees' reports on them, the conditional verdict could be upgraded. In my reading of the manuscript itself, the internal proof chain is coherent and I did not find a counterexample to the main theorems. The paper is very long, but the Polish-space section is sufficiently self-contained once the imported vague-metric facts are accepted; the applications in Section 7 are the least developed part and should be checked with particular care."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Noda proves aging and sub-aging for Bouchaud trap models on recurrent resistance metric spaces, replacing the volume-doubling condition of Croydon–Hambly–Kumagai with a non-explosion condition. That is a real advance: CHK left aging open, and the applications to critical Galton–Watson trees and critical Erdős–Rényi graphs are new. The Sierpiński gasket case is also covered. I read the proof with some care and could not find a hidden assumption or an internal contradiction. The two key steps—the Polish metrization of the vague-and-point-process topology and the transition-density precompactness—are well executed, and the deterministic trap convergence (Theorems 5.7 and 5.11) gives a crisp route to the main results. The stress-test note agrees: no load-bearing flaw found.\n\nThe soft spots are real but not fatal. The paper leans heavily on the author's own unpublished preprints [37], [38], and [39] for the Gromov–Hausdorff-type metrization, resistance-form background, and some process convergence. That is normal in a programme of papers, but it makes verification harder and should be addressed—posting the preprints or moving the necessary lemmas into the paper or an appendix. The biggest exposition gap is in Section 7.2, where the convergence of the marked measures for the random conductance model is asserted with \"it is possible to show\" rather than proved. This is especially relevant because the sub-aging limit depends on the conductance distribution, so the verification matters. These are fixable issues, not reasons to doubt the main theorems.\n\nI disagree with any reading that calls the self-citation a circularity problem. The proofs are derivations from explicit assumptions; nothing is fitted to the conclusions. The reliance on unpublished work is a verification inconvenience, not a logic flaw.\n\nWho is this for? Specialists in trap models, aging, scaling limits of random graphs, and analysis on fractals. It deserves a serious referee, and I would be comfortable seeing it in a strong probability journal after the preprint dependency is sorted out.\n\nMy recommendation: send it to a referee who knows resistance forms, and ask the author to make the preprints available or fold the key facts into the paper.","headline":"A genuinely new aging result for Bouchaud trap models on recurrent resistance metric spaces, internally consistent and likely correct; the main caveat is heavy reliance on three unpublished preprints by the same author.","tokens_in":71670,"tokens_out":2543,"would_cite":true,"duration_ms":27951,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K37","60J27","60F17","60J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that Bouchaud trap models age and sub-age on any sequence of electrical networks that converges as resistance metric spaces and satisfies a non-explosion condition.","keywords":["Bouchaud trap model","aging","sub-aging","resistance metric spaces","electrical networks","Gromov-Hausdorff-vague topology","Sierpinski gasket","critical random graphs"],"falsifier":"One concrete check: for the standard Sierpi\\'nski gasket graph sequence, evaluate the annealed two-point function at times $s(5/3)^n 3^{n/\\alpha}$ and $t(5/3)^n 3^{n/\\alpha}$; the theorem says it converges to $\\mathbb{E}[\\Phi^{\\nu}(s,t)]$ for the Poisson-trap speed measure on the gasket. A different limit, non-convergence, or a limit that the transition-density precompactness argument cannot produce would settle whether the stated assumptions are sufficient.","tokens_in":70657,"feed_emoji":"⏳","tokens_out":10339,"duration_ms":102026,"temperature":0.7,"pith_summary":"This paper proves that the Bouchaud trap model—a Markov chain on a weighted graph whose holding times at vertices are independent heavy-tailed traps—ages and sub-ages on any sequence of electrical networks that converges in the local Gromov-Hausdorff-vague topology and satisfies a non-explosion condition. The result converts the aging problem for trap models on fractal-like or random low-dimensional graphs into a statement about effective resistances. Aging means the probability that the chain is at the same vertex at two widely separated times has a nonzero limit depending only on the ratio of the times; sub-aging is the same statement on a shorter waiting-time window. The theorem covers the Sierpi\\'nski gasket, critical Galton-Watson trees, and the critical Erd\\H{o}s-R\\'enyi random graph, and it upgrades an earlier convergence result by replacing a uniform volume-doubling assumption with the weaker non-explosion condition.","feed_headline":"Trap-model aging follows from resistance convergence","feed_subtitle":"A weaker non-explosion condition unlocks aging and sub-aging on gaskets, critical trees, and random graphs.","key_machinery":"The load-bearing object is the vague-and-point-process topology on the space of discrete Radon measures: it keeps track not only of where traps sit and their total mass, but also of the individual atoms, so that the heavy-tailed trap environment converges as a marked point process. The paper proves this topology is Polish, which lets the Skorohod representation theorem produce an almost-sure coupling of the traps. On the resistance side, the key estimates are the Chapman-Kolmogorov and Cauchy-Schwarz bounds on the transition densities of the Hunt process associated with a resistance form; Proposition 4.13 shows that, along a Gromov-Hausdorff-vague convergent sequence, the transition densities are uniformly bounded and equicontinuous. This precompactness, together with trace estimates for exit times from balls, upgrades convergence of traps and processes to convergence of the (sub-)aging functions.","core_discovery":"The central discovery is that, under Assumption 1.5, the entire scaled BTM package converges: the ambient resistance metric spaces with their counting measures, the law of the scaled trap measure in the vague-and-point-process topology, the law of the scaled process started at the root, and the two-point aging function. The limit is described by a random speed measure built from a Poisson point process with intensity $\\mu(dx)\\alpha v^{-1-\\alpha}dv$ on the limiting resistance space; this is the analogue of the one-dimensional trap-model speed measure. In particular the annealed aging functions converge, $\\mathbb{E}_n[\\tilde{\\Phi}_n^{\\nu_n}(s,t)] \\to \\mathbb{E}[\\Phi^{\\nu}(s,t)]$. With the extra assumption that the marked counting measures $\\dot{\\mu}^\\#_n$—one-point local data such as total conductance or degree—converge, the same convergence holds for the sub-aging functions $\\tilde{\\Psi}_n^{\\nu_n}$. The proof reduces aging to deterministic trap convergence: once traps converge in the vague-and-point-process topology, transition-density precompactness and pointwise evaluation give the aging limit.","pith_inferences":["The non-explosion condition is likely close to optimal: any convergent sequence of recurrent networks where it fails should be a natural source of counterexamples to universal aging, since the proof's control of exit times and transition-density precompactness breaks exactly there.","The Polish metrization constructed for traps should transfer to other random-speed processes whose jump rates depend on finite local data, as the paper's Remark 1.11 hints for generalized trap models; a concrete next step would be to verify the analogous marked-measure convergence for such models.","Because the sub-aging limit depends on the marked measure $\\dot{\\mu}^{\\#}$ and not just on the base space, sub-aging offers a quantitative probe of local geometry; comparing finite-graph simulations to the stated Poisson limits on the gasket or Galton-Watson trees would be a direct numerical check.","The paper's Remark 1.18 suggests the arguments extend to non-symmetric BTMs once the associated random resistance metrics are shown to converge; verifying that condition for a natural heavy-tailed conductance model would test the scope of the approach."],"forward_implications":["On the Sierpi\\'nski gasket graph sequence, the BTM ages at time scale $(5/3)^n 3^{n/\\alpha}$, with the limit described by the trap process on the gasket.","On the critical Galton-Watson tree with $n$ vertices, the BTM ages at scale $n^{1/2} n^{1/\\alpha}$, and the annealed two-point functions converge to the continuum random tree limit.","On the largest component of the critical Erd\\H{o}s-R\\'enyi graph, the BTM ages at scale $n^{1/3} n^{2/(3\\alpha)}$, with limit space a fused tilted Brownian continuum random tree.","Under the degree-marked convergence assumption, the sub-aging functions converge too; the limit is expressed through the joint distribution of location and total conductance, so local graph structure is visible in the sub-aging limit.","For random conductance models on $\\mathbb{Z}$, aging homogenizes away the random conductances, while sub-aging retains their effect through the total-conductance distribution at vertices."],"supporting_citations":[{"why":"It supplies the non-explosion condition and the theorem that scaled processes associated with a convergent sequence of resistance forms converge, which is used throughout the proof.","marker":"[21]"},{"why":"It introduces the point-process topology and the one-dimensional aging result that Theorem 1.6 generalizes and recovers.","marker":"[25]"},{"why":"It gives the earlier BTM convergence theorem under uniform volume doubling, which the present paper improves by replacing that assumption with non-explosion.","marker":"[22]"},{"why":"It provides the resistance-form and transition-density estimates on which the precompactness arguments in Sections 4 and 5 rest.","marker":"[32]"},{"why":"It supplies the Polish metrization of Gromov-Hausdorff-type topologies used to state and prove the joint convergence of spaces, traps, processes, and aging functions.","marker":"[38]"},{"why":"It is the source of the vague convergence and Poisson random measure results used to prove distributional convergence of the trap point processes.","marker":"[28]"},{"why":"It gives the continuum limit of critical random graphs and the tilted-tree construction used for the Erd\\H{o}s-R\\'enyi application.","marker":"[2]"},{"why":"It provides the contour-process limit for conditioned Galton-Watson trees used in the critical tree application.","marker":"[24]"},{"why":"It describes the scaling limit of the largest critical Erd\\H{o}s-R\\'enyi component, which the application section combines with the fused-space theorem.","marker":"[4]"}],"fun_headline_variants":["Aging follows from resistance convergence in trap models","Convergent resistance spaces prove trap-model aging","Non-explosion plus convergence: aging for Bouchaud traps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the non-explosion condition: as the radius $r$ grows, the scaled effective resistance from the root to the complement of the ball of radius $a_n r$ must tend to infinity, uniformly in $n$; if this fails, the proof gives no uniform control on exit times or transition densities, and the aging conclusion is not established.","fun_headline_variants_meta":{"raw":{"variants":["Aging follows from resistance convergence in trap models","Convergent resistance spaces prove trap-model aging","Non-explosion plus convergence: aging for Bouchaud traps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000429,"raw_usage":{"total_tokens":2177,"prompt_tokens":911,"completion_tokens":1266,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":1216}},"tokens_in":527,"tokens_out":1266,"duration_ms":13000,"temperature":1.0,"reasoning_tokens":1216,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:03:29.537881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: for the standard Sierpi\\'nski gasket graph sequence, evaluate the annealed two-point function at times $s(5/3)^n 3^{n/\\alpha}$ and $t(5/3)^n 3^{n/\\alpha}$; the theorem says it converges to $\\mathbb{E}[\\Phi^{\\nu}(s,t)]$ for the Poisson-trap speed measure on the gasket. A different limit, non-convergence, or a limit that the transition-density precompactness argument cannot produce would settle whether the stated assumptions are sufficient.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the non-explosion condition and the theorem that scaled processes associated with a convergent sequence of resistance forms converge, which is used throughout the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces the point-process topology and the one-dimensional aging result that Theorem 1.6 generalizes and recovers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the earlier BTM convergence theorem under uniform volume doubling, which the present paper improves by replacing that assumption with non-explosion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the resistance-form and transition-density estimates on which the precompactness arguments in Sections 4 and 5 rest."},{"cited_title":"Kallenberg, Random measures, theory and applications , Probability Theory and Stochastic Mod- elling, vol","cited_arxiv_id":null,"evidence_quote":"It is the source of the vague convergence and Poisson random measure results used to prove distributional convergence of the trap point processes."},{"cited_title":"Addario-Berry, N","cited_arxiv_id":null,"evidence_quote":"It gives the continuum limit of critical random graphs and the tilted-tree construction used for the Erd\\H{o}s-R\\'enyi application."},{"cited_title":"Duquesne, A limit theorem for the contour process of conditioned Galto n-Watson trees , Ann","cited_arxiv_id":null,"evidence_quote":"It provides the contour-process limit for conditioned Galton-Watson trees used in the critical tree application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It describes the scaling limit of the largest critical Erd\\H{o}s-R\\'enyi component, which the application section combines with the fused-space theorem."}],"review_version":1}