{"id":"13e713d5-474f-45c4-b498-03f864dc42c8","arxiv_id":"2505.09318","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Clique and subtree counts in the age-dependent random connection model are asymptotically normal in the light-tailed regime, with quantitative bounds for cliques.","lead":"Counts of small shapes in a spatial network model with heavy-tailed connections are shown to become normally distributed in the regime where their fluctuations have finite variance. The paper proves this for cliques with explicit rates and for rooted subtrees qualitatively, completing the Gaussian side of a phase transition.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7(i) covariance decay is the load-bearing step; its proof absorbs logarithmic factors and imports an overlap-size reduction from [9], leaving the Cox-Grimmett bound (11) not fully self-contained.","rationale":"The reader's weakest assumption and my concern coincide: Lemma 7(i) is the load-bearing covariance estimate for Theorem 2, and it is not fully self-contained. The paper has genuine strengths: the Malliavin-Stein framework for clique counts is applied with a recent p-Poincar\\'e inequality, the variance asymptotics in Proposition 4 are laid out in detail, and the association-based blocking argument in Theorem 2 is structurally sound once the covariance decay is granted. I do not see a demonstrable mathematical error in the Section 6 case analysis; the issue is that the proof leaves the decisive exponent to a lengthy computation that suppresses logarithmic factors and depends on companion results [8,9]. Because the requested verdict is already CONDITIONAL, my read does not move it: the appropriate resolution is to condition acceptance on an independent verification of Lemma 7(i), ideally by completing the omitted casework or by confirming the overlap reduction on the torus. A negative outcome of the proposed check would elevate the concern to a correctness failure, but absent that evidence, rejecting the paper would be too strong.","tokens_in":31542,"tokens_out":26342,"duration_ms":262597,"concrete_test":"Carry out the Section 6 case analysis for a concrete tree, e.g. the directed wedge (\\ell=2), without discarding logarithmic factors: derive the exact k-dependence of Cov(T_1^{(n)},T_k^{(n)}) from (29) for Cases (I)-(V), and verify that the overlap-1 reduction imported from [9, Prop. 2.2] remains valid on the torus. In parallel, simulate the ADRCM for \\gamma=0.24 and moderately large n, estimate Cov(T_1,T_k) for k up to n/4, and fit the power-law exponent. If the derived or observed exponent is at most 2-4\\gamma-\\varepsilon for some \\varepsilon>0, then the Cox-Grimmett bound (11) can fail and Theorem 2 would need a revised covariance estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2 hinges on Lemma 7(i): the Cox-Grimmett coefficient in (10) is bounded in (11) by O(k^{-(1-2\\ell\\gamma)}) only if Cov(T_1^{(n)},T_k^{(n)}) = O(k^{-(2-2\\ell\\gamma)}). The proof in Section 6 imports two external ingredients: the reduction to overlap size 1 from [9, Prop. 2.2] and the remainder estimate R_k^{(n)} from [8, Lemma 10]. It also explicitly suppresses logarithmic factors, arguing they can be absorbed by factors u^{-\\eta\\gamma}; this is safe if only the qualitative vanishing of u_n(k) is needed, but the proof never establishes the exact exponent with a complete case-by-case computation. In particular, the decisive Cases (I)-(V) following (27) are presented with representative computations, and the final k-exponent 2-2\\ell\\gamma relies on the purely combinatorial relations (24)-(26). Near the threshold \\gamma=1/(2\\ell), the margin 1-2\\ell\\gamma is arbitrarily small, so a polynomial slow-down in the covariance decay -- for example, a missing factor k^{-\\varepsilon} -- would make the series in (11) diverge and would invalidate the associated-variable CLT in Theorem 2. This is the weakest point of the paper: not a demonstrated contradiction, but a load-bearing estimate that is not independently verified within the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies normal approximation for subgraph counts in the one-dimensional age-dependent random connection model (ADRCM). For clique counts and 0<γ<1/2, it proves univariate and multivariate quantitative CLTs with explicit rates using Trauthwein's p-Poincaré inequalities on Poisson space, together with variance/covariance asymptotics giving n^{-1}Cov(C_{n,k},C_{n,ℓ})→σ_{k,ℓ}. For rooted subtree counts of a directed tree T with ℓ leaves, it proves a qualitative CLT in the regime γ<1/(2ℓ) by decomposing the counts over spatial slabs, establishing positive association, and using the Cox–Grimmett coefficient. The main technical content is a long series of moment and covariance estimates for first- and second-order difference operators and for rooted-subtree counts, with many case splits. Theorems 1 and 2 complement the stable-limit results in the companion paper [9] and together describe a Gaussian-to-stable phase transition for these statistics.","tokens_in":31820,"tokens_out":12952,"duration_ms":128118,"significance":"If the main estimates are correct, the paper is a valuable contribution: it gives the first normal approximation results for ADRCM subgraph counts in the light-tailed regime where only (2+ε) moments are available, and the clique result is quantitative. The variance asymptotics and the explicit phase-transition picture with [9] are interesting, and the proof strategy is clearly organized, with most auxiliary lemmas proved in detail. However, the subtree CLT in Theorem 2 rests on one covariance decay estimate — Lemma 7(i) — whose proof is partly imported from companion preprints and partly presented through representative cases. Because that estimate is the load-bearing step for the Cox–Grimmett bound (11), the paper is not yet fully self-contained, and the central claim of Theorem 2 must be treated as conditional until the missing details are supplied.","major_comments":[{"comment":"The covariance decay Cov(T_1^{(n)}, T_k^{(n)}) = O(k^{-(2-2ℓγ)}) is the load-bearing estimate for Theorem 2, because it yields the Cox–Grimmett bound (11) with the positive margin 1-2ℓγ. The proof of Lemma 7(i) imports the reduction to overlap size one from [9, Proposition 2.2] and the remainder R_k^{(n)} from [8, Lemma 10], and the case analysis after (27) is not complete: the text states that Cases (II) and (IV) are 'almost identical' and omits their details, and logarithmic factors are suppressed by assertion rather than by an explicit argument. Since the margin 1-2ℓγ tends to zero at the boundary γ=1/(2ℓ), a small unverified polynomial factor in the covariance decay would destroy the summability in (11) and invalidate Theorem 2. Please provide a complete, self-contained proof of Lemma 7(i), or state Theorem 2 as conditional on the companion results with precise references to the exact statements used.","section":"Section 6, Lemma 7(i) and Eq. (11)"},{"comment":"The final exponent 2-2ℓγ in Lemma 7(i) is obtained by using the leaf-count relations (24)–(26) as equalities. The manuscript does not demonstrate that every possible configuration of the two copies T1 and T2, the common node, and the leaf sets falls into exactly one of (24)–(26), nor does it show that the omitted cases and the absorbed logarithmic factors cannot change the k-exponent. Because the estimate is used at a threshold where the margin is arbitrarily small, the proof should explicitly enumerate all combinatorial possibilities or provide a uniform argument that rules out any polynomial slow-down.","section":"Section 6, Eqs. (24)–(26) and (29)–(36)"},{"comment":"In the variance lower bound, the step 'liminf_n VarpT_1^{(n)}q ≥ ∫_0^1 μ_T(u)^2 du > 0' uses Fatou's lemma, but the convergence μ_{n,T}(u) → μ_T(u) is not proved. This convergence is plausible because for each fixed u the range of the ADRCM is finite and the torus neighborhoods stabilize as n→∞, but as written it is a gap in an argument that is needed for the linear lower bound on Var(T_{n,T}) used in Theorem 2. Please add the missing stabilization argument.","section":"Section 6, proof of Lemma 7(ii)"}],"minor_comments":[{"comment":"There is a typo in the title ('Approxima tion') and the phrase 'age-dependent random connection model' is repeated in the abstract; please edit.","section":"Title and abstract"},{"comment":"The sentence at Eq. (28)–(29) stating that the displayed integral is 'of order O(k^{-(2-2ℓγ)}) uniformly over n' appears before the proof of that bound; the proof later gives upper bounds, and the text should clarify that these are estimates rather than equalities.","section":"Section 6, proof of Lemma 7(i), display before Eq. (29)"},{"comment":"In the second displayed estimate after (39), the bound ∑_{q=ℓ_n+1}^n Cov(T_p^{(n)}, T_q^{(n)}) is written with u_n(p), but the argument of u_n should be the block distance ℓ_n+1-p; the displayed index is only correct after a reindexing. Please clarify this step to avoid a mistaken reading.","section":"Section 6, proof of Theorem 2, Eq. (39)"},{"comment":"Part (i) of Proposition 16 is cited to [9, Proposition 2.1] rather than proved in this paper; the proposition statement should explicitly say so, since this is the basis of the moment bounds used in Theorem 2.","section":"Proposition 16"},{"comment":"The statement that σ_{k,k}>0 is asserted without proof. The authors say it is 'easily established'; a short justification would improve rigor, as this positivity enters the normalization in the CLT.","section":"Theorem 1(b) and Proposition 4"}],"recommendation":"major_revision","confidential_remarks":"The paper relies substantially on two companion manuscripts, [8] and [9]. In particular, the central estimate for Theorem 2 — Lemma 7(i) — is not proved in full inside this paper. The editor may wish to verify that these companion papers are publicly accessible and that the cited propositions indeed contain all the needed statements. The stress-test concern about the covariance decay near the threshold is real; it is not a demonstrated contradiction, but it is a load-bearing point that deserves complete verification before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The first thing to know: this is the Gaussian half of the phase transition for clique and subtree counts in the age-dependent random connection model, and it is a real contribution. Theorem 1 is the strongest part — a quantitative multivariate CLT for cliques in the light-tailed regime, with explicit Wasserstein rates and variance asymptotics, obtained via Trauthwein's p-Poincare inequalities and a careful configurational moment analysis. Propositions 12 and 13, which control fractional moments of first- and second-order difference operators, look like the genuine new work, and they read as sound.\n\nTheorem 2, the subtree CLT, is a qualitative result (no rate), and it goes through the Cox-Grimmett route for associated variables. The proof is honest about this.\n\nNow the soft spots, in proportion. The load-bearing estimate is Lemma 7(i), the covariance decay Cov(T_1,T_k) = O(k^{-(2-2ell gamma)}). Everything in Theorem 2 depends on it. The proof is a long case analysis, but it is not fully self-contained: it imports the reduction to overlap size 1 from the companion preprint [9], imports the remainder estimate from [8, Lemma 10], suppresses logarithmic factors with an absorption argument, and leaves Cases (II) and (IV) 'omitted'. Near the threshold gamma = 1/(2ell), the margin in the Cox-Grimmett summability is arbitrarily small, so even a tiny unaccounted polynomial slow-down would break the argument. This is not a demonstrated error — the exponent is plausible and the cases that are written out check out — but it is exactly where a referee should dig.\n\nThe dependence on [9] is also worth flagging: Proposition 16(i) is quoted from the companion paper, so Theorem 2 cannot be judged without it. For a standalone paper, that is a weakness, though standard in this line of work.\n\nWho is this for: specialists in spatial random graphs and Poisson functionals. It deserves a serious referee — conditional accept, I think, with the request to make Lemma 7 self-contained and to verify the exponent in the borderline regime. If that holds, this will be a standard reference for the Gaussian side of the ADRCM.","headline":"Strong technical paper giving the Gaussian side of the phase transition for subgraph counts in the ADRCM; the clique CLT is solid, but the subtree CLT rests on a covariance decay lemma whose proof is not fully self-contained.","tokens_in":32362,"tokens_out":2966,"would_cite":true,"duration_ms":29135,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60D05","05C80","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"In the age-dependent random connection model, normalized clique and directed-tree counts are asymptotically normal whenever the typical degree has finite second moments, with explicit Wasserstein rates for cliques and a sharp leaf-count…","keywords":["age-dependent random connection model","normal approximation","central limit theorem","clique counts","subtree counts","associated random variables","Malliavin-Stein method","scale-free random networks"],"falsifier":"Simulate or compute $\\mathrm{Cov}(T_1^{(n)},T_k^{(n)})$ for a directed tree with $\\ell=2$ leaves at $\\gamma$ slightly below $1/4$: Lemma 7(i) requires a tail of order $k^{-(2-2\\ell\\gamma)}$ up to logarithmic factors, so a Monte Carlo estimate showing decay $\\Omega(k^{-(2-2\\ell\\gamma)+\\delta})$ for some $\\delta>0$ at large $k$ and $n$ would falsify the lemma, break the summability of the coefficient in (11), and remove the foundation of Theorem 2, while decay at least that fast would support the exponent the proof relies on.","tokens_in":31328,"feed_emoji":"📈","tokens_out":17886,"duration_ms":153709,"temperature":0.7,"pith_summary":"This paper proves that two families of subgraph counts in the age-dependent random connection model — a spatial network with power-law degree distribution in which points carry random birth times and connection probability depends on distance and birth times — are asymptotically normal. For clique counts it establishes a quantitative central limit theorem: after centering and scaling, $k$-clique counts converge to the standard normal in Wasserstein distance at rate $O(n^{-(\\eta-1)/2})$, and the variance grows linearly with an explicitly identified limit $\\sigma_k>0$; a multivariate version treats several clique sizes jointly. For directed-tree counts it proves distributional convergence to normality whenever the tree has $\\ell$ leaves and the tail exponent satisfies $0<\\gamma<1/(2\\ell)$. If the results are right, they justify statistical use of the model — confidence intervals and hypothesis tests for network summaries — and, together with the companion stable-limit results, they locate the transition where fluctuations switch from Gaussian to stable at $\\gamma=1/2$ for cliques and $\\gamma=1/(2\\ell)$ for trees.","feed_headline":"Spatial scale-free networks: subgraph counts go Gaussian","feed_subtitle":"Finite second moments of the degree force small-subgraph counts to be Gaussian, with explicit rates for cliques.","key_machinery":"For clique counts the engine is the Malliavin–Stein method adapted to Poisson functionals whose fourth moments are not uniformly bounded: first- and second-order difference operators $D_q C_{n,k}$ and $D_{u,q} C_{n,k}$ are controlled by moment bounds such as $\\mathbb{E}[(D_{u,q}C_{n,k})^a]\\le C(1_{\\{q\\in N^{\\mathrm{up}}(u)\\}}+v^{-a\\gamma}\\hat{s}(u,|y|)^{a\\gamma})$ with $\\hat{s}(u,r)=1\\wedge(u^\\gamma r)^{-1/(1-\\gamma)}1_{\\{r\\le 2u^{-1}\\}}$, reflecting the asymmetric structure in which the up-neighbourhood of a point with mark $u$ is Poisson of parameter of order $u^{-\\gamma}$ while the down-neighbourhood has bounded mean. Covariance asymptotics come from a Mecke-formula decomposition into one-vertex and two-vertex contributions, with torus-to-line boundary corrections controlled to order $n^{-\\zeta(\\gamma)}$. For subtree counts the machinery is different: the count is split into spatially indexed slab variables $T_i^{(n)}$, which are positively associated by the Harris–FKG inequality, and the proof reduces to summability of the Cox–Grimmett coefficient $u_n(k)$; the key input is the decay $\\mathrm{Cov}(T_1^{(n)},T_k^{(n)})=O(k^{-(2-2\\ell\\gamma)})$ derived in Section 6 from combinatorial leaf-count relations (24)–(26). Newman's inequality then bounds the characteristic-function error, and a Lindeberg check uses the moment bounds of Proposition 16.","core_discovery":"The central claim is that in the regime where a typical vertex has a neighbourhood of finite $(2+\\varepsilon)$-th moment, subgraph fluctuations occur at scale $\\sqrt{n}$ and are Gaussian with explicit limiting variances. Theorem 1 states that for $0<\\gamma<1/2$, each normalized clique count $\\widehat{C}_{n,k}$ converges to $N(0,1)$, with $|n^{-1}\\mathrm{Var}(C_{n,k})-\\sigma_k|=O(n^{2\\gamma-1})$ for an explicit $\\sigma_k>0$; the multivariate version gives joint convergence to $N(0,\\Sigma)$ with rate $O(n^{-(\\zeta(\\gamma)\\wedge(\\eta-1)/2)})$ where $\\zeta(\\gamma)=(1-\\gamma)\\wedge(1-2\\gamma)/\\gamma$. Theorem 2 states that for a directed tree $T$ with $\\ell$ leaves, whenever $0<\\gamma<1/(2\\ell)$, the normalized subtree count converges in distribution to $N(0,1)$ and its variance is of exact linear order in $n$. The thresholds are not artifacts: above $\\gamma=1/2$ for cliques and $\\gamma=1/(2\\ell)$ for trees, the same statistics obey stable limit theorems, so the two results together describe a Gaussian-to-stable phase transition.","pith_inferences":["The covariance-decay exponent $2-2\\ell\\gamma$ is forced by leaf bookkeeping rather than by spatial dimension, so the threshold $\\gamma=1/(2\\ell)$ and the subtree CLT are plausible in dimensions $d>1$ and for unbounded-support profiles as well; the paper itself flags these extensions as open in Remark 3, which makes them a concrete testable route.","The rate $O(n^{-(\\eta-1)/2})$ in Theorem 1 is tied to the auxiliary parameter $\\eta$ whose admissible range $\\eta(2\\gamma\\vee(1-\\gamma))<1$ shrinks as $\\gamma$ grows; in the sub-regime $\\gamma<1/4$ the paper suspects sharper rates exist, and the variance-error term $n^{2\\gamma-1}$ is a plausible bottleneck worth optimizing against the constants of Proposition 5.","The paper proves no rate in Theorem 2, and a quantitative subtree CLT would have to turn the covariance decay of Lemma 7(i) into an explicit block-size choice; the block argument in Section 6 shows any such rate would be governed by how rapidly the block length can grow against the $(2+\\delta)$-moment bound of Proposition 16.","A Monte Carlo study of the clustering coefficient ratio in the light-tailed regime could test the Cauchy-type conjecture directly; a visible risk is that the denominator's fluctuations are not bounded away from zero, so heavy tails of the ratio may appear even where each component is Gaussian."],"forward_implications":["For $0<\\gamma<1/2$, clique counts in the ADRCM fluctuate normally at scale $\\sqrt{n}$ with explicitly computed limiting variances $\\sigma_k>0$, and joint convergence holds for cliques of several sizes with the covariance matrix identified in Proposition 4.","Directed-tree counts are normal whenever $0<\\gamma<1/(2\\ell)$ for a tree with $\\ell$ leaves, with variance of exact linear order in the domain size, and no uniform fourth-moment assumption is needed.","Together with the companion stable-limit paper, the results locate a Gaussian-to-stable phase transition for these statistics: at $\\gamma=1/2$ for cliques and at $\\gamma=1/(2\\ell)$ for trees.","The normality results supply the theoretical basis for statistical inference with the ADRCM, namely asymptotic confidence intervals and hypothesis tests for subgraph-based summaries such as wedge and triangle counts.","The paper conjectures that the clustering coefficient — the ratio of wedge counts to triangle counts — converges to a Cauchy-type limit of the form $a+bZ_1/Z_2$ with $Z_1,Z_2$ independent standard normals in the light-tailed regime."],"supporting_citations":[{"why":"Supplies the univariate quantitative CLT on Poisson space (Skorohod estimates and p-Poincaré inequalities) that converts the Gamma_i bounds into the Wasserstein rate of Theorem 1(a).","marker":"[26]"},{"why":"Supplies the multivariate second-order p-Poincaré inequality and the C3-based distance used in the multivariate statement of Theorem 1(b).","marker":"[25]"},{"why":"Supplies the central limit theorems for associated random variables (Theorems 4.1 and 4.8) that yield the subtree CLT of Theorem 2.","marker":"[19]"},{"why":"Defines the Cox–Grimmett coefficient u_n(k) whose summability is the pivotal condition of the subtree proof.","marker":"[3]"},{"why":"Provides the Poisson up- and down-neighbourhood size result reproduced as Lemma 8, which drives all the moment bounds for clique counts.","marker":"[6]"},{"why":"Companion paper establishing stable limit theorems above the thresholds and supplying moment and covariance estimates for subtree and clique counts that the present proofs reuse.","marker":"[9]"},{"why":"Source of the Harris–FKG inequality used to establish positive association of the slab variables T_i^{(n)}.","marker":"[16]"},{"why":"Newman's inequality, stated as Theorem 18, is used to bound the characteristic-function error in the block argument for Theorem 2.","marker":"[17]"}],"fun_headline_variants":["Subgraph counts go Gaussian in scale-free networks","Clique and tree counts: Gaussian limits with explicit rates","Phase transition: subgraph counts switch from Gaussian to stable","Age-dependent random connection model: subgraph normality","Heavy tails to Gaussian: subgraph counts in random networks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The subtree result stands on the claim, proved by the case analysis of Section 6, that the covariance between rooted subtree counts in torus slabs separated by $k$ decays like $k^{-(2-2\\ell\\gamma)}$ up to logarithmic factors, which the proof discards; if the true decay were slower than that by more than a logarithmic factor near the threshold $\\gamma=1/(2\\ell)$, the Cox–Grimmett coefficients in (11) would cease to be summable and the associated-variable argument for Theorem 2 would break.","fun_headline_variants_meta":{"raw":{"variants":["Subgraph counts go Gaussian in scale-free networks","Clique and tree counts: Gaussian limits with explicit rates","Phase transition: subgraph counts switch from Gaussian to stable","Age-dependent random connection model: subgraph normality","Heavy tails to Gaussian: subgraph counts in random networks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000602,"raw_usage":{"total_tokens":2785,"prompt_tokens":896,"completion_tokens":1889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":1811}},"tokens_in":512,"tokens_out":1889,"duration_ms":12877,"temperature":1.0,"reasoning_tokens":1811,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:35:22.094586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate or compute $\\mathrm{Cov}(T_1^{(n)},T_k^{(n)})$ for a directed tree with $\\ell=2$ leaves at $\\gamma$ slightly below $1/4$: Lemma 7(i) requires a tail of order $k^{-(2-2\\ell\\gamma)}$ up to logarithmic factors, so a Monte Carlo estimate showing decay $\\Omega(k^{-(2-2\\ell\\gamma)+\\delta})$ for some $\\delta>0$ at large $k$ and $n$ would falsify the lemma, break the summability of the coefficient in (11), and remove the foundation of Theorem 2, while decay at least that fast would support the exponent the proof relies on.","supporting_citations":[{"cited_title":"Trauthwein","cited_arxiv_id":null,"evidence_quote":"Supplies the univariate quantitative CLT on Poisson space (Skorohod estimates and p-Poincaré inequalities) that converts the Gamma_i bounds into the Wasserstein rate of Theorem 1(a)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the central limit theorems for associated random variables (Theorems 4.1 and 4.8) that yield the subtree CLT of Theorem 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Cox–Grimmett coefficient u_n(k) whose summability is the pivotal condition of the subtree proof."},{"cited_title":"Gracar, L","cited_arxiv_id":null,"evidence_quote":"Provides the Poisson up- and down-neighbourhood size result reproduced as Lemma 8, which drives all the moment bounds for clique counts."},{"cited_title":"Limit theorems under heavy-tailed scenario in the age dependent random connection models","cited_arxiv_id":"2409.05226","evidence_quote":"Companion paper establishing stable limit theorems above the thresholds and supplying moment and covariance estimates for subtree and clique counts that the present proofs reuse."},{"cited_title":"Last and M","cited_arxiv_id":null,"evidence_quote":"Source of the Harris–FKG inequality used to establish positive association of the slab variables T_i^{(n)}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Newman's inequality, stated as Theorem 18, is used to bound the characteristic-function error in the block argument for Theorem 2."}],"review_version":1}