{"id":"d6d25be9-9762-4b0c-a093-c8e08a2f5863","arxiv_id":"2507.12502","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper claims a quantitative Berry-Esseen bound for edge eigenvectors of random regular graphs, but the proof relies on an incorrect local law and contradicts itself on the rate.","lead":"This paper claims the first explicit convergence rate for how quickly edge eigenvector statistics of random regular graphs become Gaussian, with a bound of N^{-1/6+ε}. It introduces a constrained Dyson Brownian motion to compare the graph to a Gaussian ensemble, but the proof contains internal contradictions and a likely false local law.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For fixed d the edge resolvent limit is Kesten–McKay, not semicircle; Theorem 4.3's N^{-5/6} local law is off by O(1), sinking the Berry–Esseen bound.","rationale":"Reading in good faith, the paper's central claim is a quantitative Berry–Esseen bound for edge eigenvector overlaps on fixed-degree random regular graphs. The argument hinges entirely on the sharp edge isotropic local law, Theorem 4.3, which controls resolvent entries to N^{−5/6+ε} around the semicircle value. This is the load-bearing condition: the overlap SDE error term, moment evolution, cumulant comparison, and backward propagation all depend on it. The concern is not about the constants or the rate being beyond current consensus; it is an internal mathematical inconsistency. For H̃, the sequence of graphs has an empirical spectral measure converging to the Kesten–McKay law, whose Stieltjes transform differs from the semicircle transform by an O(1) amount at the spectral edge for fixed d. At η = N^{−2/3}, the claimed error N^{−5/6+ε} is far smaller than this deterministic gap. Moreover, the self-consistent equation written in Eq. (4.5) does not reduce to the semicircle equation in any limit, since it contains a (d−1) factor that changes the edge location. Thus the proof of Theorem 4.3 cannot be repaired by adjusting constants; the object being controlled is not the deterministic limit of the resolvent. Because this local law feeds every subsequent estimate, the main theorem is unsupported. The reader's weakest assumption identified exactly this semicircle-versus-Kesten–McKay mismatch, and I agree. The secondary contradiction between Theorem 2.3 and Corollary 2.4 about the same CDF further weakens the presentation, but the local law failure alone is decisive.","tokens_in":29561,"tokens_out":17400,"duration_ms":167571,"concrete_test":"Take d=3. Compute the Kesten–McKay Stieltjes transform for H̃ exactly: m_KM(z) = −(d−1)[(d−2)z − d√(z²−4)]/[2(d²−(d−1)z²)]. Evaluate at z = 2 + iN^{−2/3} with N = 10^6 and compare with m_sc(z) = (−z+√(z²−4))/2. If |m_KM − m_sc| is bounded below by a constant near 1/(d−2) = 1 while N^{−5/6} ≈ 10^{−5}, Theorem 4.3 fails. Also substitute m_sc into Eq. (4.5); the residual does not vanish as η → 0, confirming the proof's self-consistent equation is not satisfied by the claimed limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For fixed d ≥ 3, the deterministic limit of the resolvent of H̃ = A/√(d−1) − (d/√(d−1))ee^T/N is the Kesten–McKay Stieltjes transform m_KM, not the semicircle m_sc. Theorem 4.3 claims |⟨q,G(z)q⟩ − m_sc(z)| ≤ C N^{−5/6+ε} for z near the edge, but at z = 2+iη with η = N^{−2/3}, the real parts satisfy m_KM − m_sc → −1/(d−2), an O(1) difference, while N^{−5/6} → 0. Hence the asserted local law is false for every fixed d. The proof is internally inconsistent: Eq. (4.5), Gqq = −1/(z + (d−1)Gqq), is the self-consistent equation for support [−2√(d−1), 2√(d−1)] (variance d−1), not the semicircle equation m_sc = −1/(z + m_sc) satisfied by the stated m_sc. Consequently the overlap SDE error bound (Prop. 4.2) and moment evolution (Thm. 4.5), which both invoke Thm. 4.3, inherit this failure, and the final N^{−1/6+ε} Berry–Esseen bound (Thm. 2.3) does not follow. Secondary inconsistency: Thm. 2.3 and Cor. 2.4 state different N exponents for the same CDF supremum, with Remark 2.5 claiming Thm. 2.3 applies only to smooth test functions despite its CDF statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims quantitative Berry-Esseen bounds for edge eigenvector overlaps in fixed-degree random regular graphs. The central object is the normalized overlap sqrt(N)<q,u_2> for a deterministic unit vector q orthogonal to the all-ones vector, for which the paper states sup_x |P(sqrt(N)<q,u_2> <= x) - Phi(x)| <= C_d N^{-1/6+epsilon}. The proof proceeds through a constrained Dyson Brownian motion that preserves the degree constraint, a sharp edge isotropic local law for the resolvent near the spectral edge, a moment evolution argument, a fourth-order cumulant comparison with constrained GOE, and a backward stability estimate. The paper also claims a joint central limit theorem for several top eigenvectors and gives heuristic evidence that the N^{-1/6} rate is optimal. The claimed contributions are strong and quantitative, but the manuscript contains a direct contradiction between the main theorem and its corollary, and the key edge isotropic local law is derived from the wrong limiting law for fixed-degree regular graphs.","tokens_in":29937,"tokens_out":5183,"duration_ms":61114,"significance":"If the results were correct, this would be a significant advance: explicit rates and constants for edge eigenvector universality would complement the qualitative results of He-Huang-Yau and would provide finite-size tools for spectral algorithms. The paper is also commendable for attempting to track all constants explicitly and for proposing a single-scale comparison method. However, the two most load-bearing components are inconsistent or incorrect: the main theorem and its corollary give different rates for the same cumulative distribution function, and the sharp edge local law compares the resolvent against the semicircle law instead of the Kesten-McKay law appropriate for fixed d-regular graphs. Because these issues are central, the claimed results are not supported by the manuscript.","major_comments":[{"comment":"Theorem 2.3 and Corollary 2.4 state different rates for the identical object: Theorem 2.3 asserts sup_x |P(sqrt(N)<q,u_2> <= x) - Phi(x)| <= C_d N^{-1/6+epsilon}, while Corollary 2.4 asserts the same supremum is <= C_d N^{-5/36+epsilon}. Remark 2.5 then says that Theorem 2.3 applies only to smooth test functions, although both the theorem and the abstract state a bound for the cumulative distribution function. Since the proofs in Section 7.2 and Appendix B lead to these different rates, the main quantitative claim is internally inconsistent and needs to be resolved before the paper can be evaluated.","section":"Section 2.3 (Theorem 2.3, Corollary 2.4, Remark 2.5)"},{"comment":"The sharp edge isotropic local law compares <q,G(z)q> with the semicircle Stieltjes transform m_sc, but for a fixed-degree d-regular graph the correct deterministic limit is the Kesten-McKay transform, not m_sc. Moreover, Eq. (4.5) states G_qq(z) = -1/(z + (d-1)G_qq(z)), which is the self-consistent equation for the Kesten-McKay law with variance parameter d-1, not the semicircle equation m_sc = -1/(z + m_sc). Near the edge, for z = 2 + i eta with eta = N^{-2/3}, the difference m_KM(z) - m_sc(z) tends to an O(1) constant, whereas the claimed right-hand side N^{-5/6+epsilon} tends to zero. Thus Theorem 4.3 is false as stated for every fixed d. Since Proposition 4.2, Theorem 4.5, and the backward stability argument in Section 7 all invoke Theorem 4.3, the final Berry-Esseen bound does not follow.","section":"Section 4.2, Eq. (4.5), Theorem 4.3"},{"comment":"The proof of Theorem 6.5 chooses delta = N^{-1/10} and obtains an intermediate error of C N^{-1/3+2epsilon} + C N^{-1/10}, but the displayed statement then drops the N^{-1/10} term and claims an error C_GOE N^{-1/2} + C N^{-1/3+2epsilon}. The dropped term is larger than the N^{-1/3+2epsilon} term and is also larger than the N^{-1/6+3epsilon} backward stability bound used in the proof of Theorem 2.3, so the advertised rate is not derived from the given estimates.","section":"Section 6.3, proof of Theorem 6.5"},{"comment":"The proof of Lemma 6.4 applies the classical Berry-Esseen theorem for sums of independent random variables to the GOE eigenvector overlap X = sqrt(N)<q,v_2^W>, although the eigenvector components are not independent. The subsequent claims about the Lyapunov ratio and the O(N^{-1/2}) rate are asserted rather than proved; the cited references may contain such quantitative eigenvector CLT results, but the derivation presented here does not support the stated bound.","section":"Section 6.3, Lemma 6.4"}],"minor_comments":[{"comment":"The abstract says the theorem holds for any d-regular graph on N vertices, while Theorem 2.3 states that G is a uniformly random d-regular graph; the deterministic statement is different from the probabilistic one and should be clarified.","section":"Abstract and Section 2.3"},{"comment":"The informal statement says indicator functions achieve the rate N^{-5/36+epsilon}, but the abstract and Theorem 2.3 present N^{-1/6+epsilon} for the CDF; this inconsistency should be resolved in favor of a single rate for the CDF supremum.","section":"Section 1.2"},{"comment":"The example calculation for a 3-regular graph with N = 10^6 contains a consistency issue: the displayed numerical bound is about 0.115 C_3, but the text in Section 1.2 suggests the statistics are within 0.05 of Gaussian for that size, and the two statements are not reconciled with the given constants.","section":"Section 3.3"}],"recommendation":"reject","confidential_remarks":"The manuscript has a direct internal contradiction concerning the claimed rate for the same CDF object, and the central local law is based on the wrong limiting law for fixed-degree regular graphs. These are not presentation issues; they invalidate the main theorem. I do not see a fix that stays within the scope of the manuscript, so rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the main theorem is not proven, and the central edge local law is false as stated. The author is aiming at a real gap—no explicit Berry-Esseen rate exists for edge eigenvectors of fixed-degree random regular graphs—and the single-scale constrained DBM comparison is a reasonable idea. The constant tracking is honest in spirit. But the core technical component, Theorem 4.3, is built on the wrong limiting law. For fixed d≥3 the resolvent near the edge converges to the Kesten–McKay law, not the semicircle law. Equation (4.5) even writes the self-consistent equation with (d−1), which is the Kesten–McKay equation, not the semicircle equation m_sc = −1/(z+m_sc). Near z = 2+iN^{−2/3} the two Stieltjes transforms differ by an O(1) constant, while N^{−5/6} goes to zero. So the claimed N^{−5/6} local law is false, and the overlap SDE error, moment evolution, and cumulant comparison all inherit the failure.\n\nThere is also an internal contradiction visible without any deep computation. Theorem 2.3 states a CDF bound at rate N^{−1/6+ε}; Corollary 2.4 states the same supremum for the same CDF at rate N^{−5/36+ε}; Remark 2.5 tries to reconcile this by saying Theorem 2.3 applies only to smooth test functions, but the theorem is stated as a CDF supremum. The abstract repeats N^{−1/6}. You cannot have two different rates for the same object.\n\nTwo minor points: the claim that HHY's proof implicitly contains the N^{−1/6} rate is speculation—HHY give no rate. Section 8's optimality discussion is heuristic and does not compensate for the failed local law.\n\nWho is this for: someone tracking quantitative edge universality could read it as a cautionary example, but it should not be cited as a source of theorems. I would desk reject it, not send it out for a full referee cycle. The contradictions and the wrong local law need to be fixed before any part of this is credible.","headline":"The claimed Berry-Esseen bound rests on an edge local law built from the wrong Stieltjes transform, and Theorem 2.3 contradicts Corollary 2.4 on the same CDF.","tokens_in":30413,"tokens_out":3387,"would_cite":false,"duration_ms":38169,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","05C80","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a fixed-degree random regular graph, the normalized projection of the second eigenvector onto any fixed direction is Gaussian up to a tracked $N^{-1/6+\\varepsilon}$ error, proved with explicit constants.","keywords":["random regular graphs","edge eigenvector universality","Berry-Esseen bounds","Dyson Brownian motion","isotropic local law","semicircle law","eigenvector delocalization","spectral algorithms"],"falsifier":"Numerically evaluate $\\langle\\mathbf{q},(\\tilde{H}-(2+\\mathrm{i}N^{-2/3}))^{-1}\\mathbf{q}\\rangle$ for a random 3-regular graph at large $N$, with $\\mathbf{q}$ a fixed unit vector orthogonal to the all-ones vector, and compare with $m_{\\mathrm{sc}}(2+\\mathrm{i}N^{-2/3})$; the paper's chain requires this difference to shrink like $N^{-5/6+\\varepsilon}$, whereas the fixed-degree edge limit differs from the semicircle value by an $O(1)$ constant, so an observed $O(1)$ or even $N^{-1/3}$ discrepancy would refute the sharp local law that the bound depends on.","tokens_in":29341,"feed_emoji":"🎲","tokens_out":14480,"duration_ms":141017,"temperature":0.7,"pith_summary":"This paper tries to establish a quantitative version of edge eigenvector universality for sparse regular graphs. For a uniformly random $d$-regular graph on $N$ vertices with fixed $d\\ge 3$ — every vertex has degree exactly $d$ — and any deterministic unit vector $\\mathbf{q}$ orthogonal to the all-ones vector, the overlap $\\sqrt{N}\\langle\\mathbf{q},\\mathbf{u}_2\\rangle$ of the second eigenvector with $\\mathbf{q}$ has a distribution function that is within $C_d N^{-1/6+\\varepsilon}$ of the standard normal, uniformly in $x$. If true, this is the first explicit convergence rate for edge eigenvector statistics in this setting, with the constant $C_d\\le\\tilde{C}d^3\\varepsilon^{-10}$ tracked through the proof. The payoff is finite-size applicability: spectral clustering, network centrality, and delocalization arguments on concrete graphs need quantitative bounds rather than only a limiting statement. The paper also proves joint convergence of the top $K$ edge eigenvectors to independent Gaussians for $K\\le N^{1/10-\\delta}$, and gives evidence that $N^{-1/6}$ is the best rate available.","feed_headline":"Edge eigenvectors of random regular graphs are Gaussian to N^{-1/6}","feed_subtitle":"This is the first explicit convergence rate for sparse edge eigenvectors, with constants tracked for finite-size algorithms.","key_machinery":"The load-bearing mechanism is constrained Dyson Brownian motion, the flow $\\mathrm{d}\\tilde{H}_t=-\\tfrac12\\tilde{H}_t\\,\\mathrm{d}t+N^{-1/2}\\mathrm{d}W_t$ on symmetric matrices with zero row sums, so $\\tilde{H}_t\\mathbf{e}=0$ at every time. The overlap processes $X_i^{(q)}(t)=\\sqrt{N}\\langle\\mathbf{q},\\mathbf{u}_i(t)\\rangle$ obey an SDE whose error term is controlled by a sharp edge isotropic local law: for $z=E+\\mathrm{i}\\eta$ with $|E-2|\\le N^{-2/3+\\varepsilon}$ and $\\eta\\ge N^{-2/3}$, the claim is $|\\langle\\mathbf{q},(\\tilde{H}_t-z)^{-1}\\mathbf{q}\\rangle-m_{\\mathrm{sc}}(z)|\\le C(d,\\varepsilon)N^{-5/6+\\varepsilon}$, where $m_{\\mathrm{sc}}$ is the Stieltjes transform of the semicircle law. That local law feeds the second- and fourth-moment evolution of overlaps, the decorrelation estimates between different eigenvectors, and the fourth-order cumulant comparison with constrained GOE at the critical time. A time-reversed diffusion estimate bounds the change in expectation between time zero and $t_*$, and that backward step is where the final $N^{-1/6}$ loss enters.","core_discovery":"The central claim is Theorem 2.3: for any fixed-degree random regular graph and any fixed direction $\\mathbf{q}\\perp\\mathbf{e}$, the cumulative distribution function of $\\sqrt{N}\\langle\\mathbf{q},\\mathbf{u}_2\\rangle$ satisfies $\\sup_x|\\mathbb{P}(\\sqrt{N}\\langle\\mathbf{q},\\mathbf{u}_2\\rangle\\le x)-\\Phi(x)|\\le C_d N^{-1/6+\\varepsilon}$ with $C_d\\le\\tilde{C}d^3\\varepsilon^{-10}$. The proof flows the adjacency matrix through a degree-constrained Ornstein-Uhlenbeck process, compares once at the critical time $t_*=N^{-1/3+\\varepsilon}$ with a constrained Gaussian orthogonal ensemble using a fourth-order cumulant expansion, and then propagates the comparison backward to the original graph. A second theorem gives joint universality: the projections of the top $K$ edge eigenvectors onto any finite collection of test vectors converge to independent standard Gaussians for $K\\le N^{1/10-\\delta}$, with multivariate Berry-Esseen rate $C_{d,m}K^{3/2}N^{-1/6+\\varepsilon}$ over convex sets. Because the distribution-function statement requires smoothing an indicator, the paper's corollary for cumulative distribution functions carries the weaker rate $N^{-5/36+\\varepsilon}$.","pith_inferences":["The paper states the sharp local law against the semicircle Stieltjes transform, but for fixed $d$ the natural edge limit is the fixed-degree resolvent, which differs from semicircle by an $O(1)$ constant at $E=2$; if that discrepancy is not cancelled, the claimed $N^{-5/6+\\varepsilon}$ bound would need repair. This is an editorial caution, not a claim of the paper.","The same constraint-preserving single-scale scheme should carry over to other ensembles with a linear invariant — random lifts, graphs with fixed degree sequence — and would predict the same $N^{-1/6}$-type Berry-Esseen rate there.","The stated bound $K\\le N^{1/10-\\delta}$ is called technical in the paper; a direct numerical measurement of the covariance matrix of the top $K$ overlaps for $K$ in $[N^{1/10},N^{1/3}]$ would show whether the decorrelation mechanism genuinely degrades at that scale.","If $N^{-1/6}$ is a true barrier, spectral algorithms whose outputs are functions of edge eigenvector projections should show fluctuations of order $N^{-1/6}$ at finite $N$; that is testable by simulation of clustering or embedding errors."],"forward_implications":["For any fixed $d$, the projection of the second eigenvector onto a fixed direction is Gaussian up to an explicit $N^{-1/6+\\varepsilon}$ error, so eigenvector statistics on finite graphs come with concrete bounds rather than only as limits.","The top $K$ edge eigenvectors are jointly Gaussian and mutually independent in the limit for $K\\le N^{1/10-\\delta}$, giving a quantitative foundation for multi-dimensional spectral embeddings and clustering.","The Berry-Esseen bound implies quantitative delocalization: with high probability $\\|\\mathbf{u}_2\\|_\\infty\\le C\\sqrt{\\log N}/\\sqrt{N}$, and the mass of $\\mathbf{u}_2$ on large subsets is controlled up to explicit error.","Spectral embeddings built from $K$ edge eigenvectors separate large vertex sets by $\\Omega(1/\\sqrt{K})$ with high probability, the paper's stated justification for spectral clustering on sparse regular graphs.","The paper's optimality analysis, from edge spacing $\\Theta(N^{-2/3})$, the minimal mixing time $t_*\\sim N^{-1/3}$, and consistency across methods, predicts that no dynamical proof can beat the $N^{-1/6}$ rate for fixed-degree regular graphs."],"supporting_citations":[{"why":"The qualitative edge eigenvector universality result whose convergence rate this paper makes explicit.","marker":"[26]"},{"why":"Gives the edge eigenvalue location $\\lambda_2=2+O(N^{-2/3+o(1)})$ used throughout the rigidity and spacing estimates.","marker":"[22]"},{"why":"Origin of the Brownian-motion eigenvalue model that the constrained flow generalizes.","marker":"[16]"},{"why":"Supplies the multivariate Berry-Esseen theorem used to get the convex-set rate in the joint CLT.","marker":"[4]"},{"why":"Supplies the local semicircle law used in the resolvent correction estimates for the overlap SDE.","marker":"[20]"},{"why":"Supplies the sparse-matrix local law and edge rigidity used near the spectral edge.","marker":"[37]"},{"why":"Provides eigenvalue rigidity and eigenvector moment bounds used in the GOE comparison.","marker":"[19]"},{"why":"Provides the eigenvector distribution results behind the GOE Berry-Esseen bound for overlaps.","marker":"[36]"},{"why":"Supplies the time-reversal theory for diffusions that underlies the backward stability estimate.","marker":"[25]"}],"fun_headline_variants":["Edge eigenvectors of random regular graphs: explicit Gaussian rate N^{-1/6}","First explicit Berry-Esseen rate for edge eigenvectors: N^{-1/6}","Edge eigenvector Gaussianity: explicit rate N^{-1/6} for random regular graphs","Random regular graph edge eigenvectors: Gaussian with explicit N^{-1/6} rate","Quantitative edge eigenvector universality: explicit rate N^{-1/6}"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the sharp edge isotropic local law — that near $E=2$, the resolvent entry $\\langle\\mathbf{q},(\\tilde{H}-z)^{-1}\\mathbf{q}\\rangle$ is within $N^{-5/6+\\varepsilon}$ of the semicircle Stieltjes transform at $z=2+\\mathrm{i}N^{-2/3}$ — because the overlap SDE errors, moment evolution, and cumulant comparison all inherit this bound; if that comparison fails, the final rate collapses.","fun_headline_variants_meta":{"raw":{"variants":["Edge eigenvectors of random regular graphs: explicit Gaussian rate N^{-1/6}","First explicit Berry-Esseen rate for edge eigenvectors: N^{-1/6}","Edge eigenvector Gaussianity: explicit rate N^{-1/6} for random regular graphs","Random regular graph edge eigenvectors: Gaussian with explicit N^{-1/6} rate","Quantitative edge eigenvector universality: explicit rate N^{-1/6}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0011,"raw_usage":{"total_tokens":4723,"prompt_tokens":1214,"completion_tokens":3509,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":830,"completion_tokens_details":{"reasoning_tokens":3401}},"tokens_in":830,"tokens_out":3509,"duration_ms":28994,"temperature":1.0,"reasoning_tokens":3401,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:58:47.801529+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluate $\\langle\\mathbf{q},(\\tilde{H}-(2+\\mathrm{i}N^{-2/3}))^{-1}\\mathbf{q}\\rangle$ for a random 3-regular graph at large $N$, with $\\mathbf{q}$ a fixed unit vector orthogonal to the all-ones vector, and compare with $m_{\\mathrm{sc}}(2+\\mathrm{i}N^{-2/3})$; the paper's chain requires this difference to shrink like $N^{-5/6+\\varepsilon}$, whereas the fixed-degree edge limit differs from the semicircle value by an $O(1)$ constant, so an observed $O(1)$ or even $N^{-1/3}$ discrepancy would refute the sharp local law that the bound depends on.","supporting_citations":[{"cited_title":"Gaussian Waves and Edge Eigenvectors of Random Regular Graphs","cited_arxiv_id":"2502.08897","evidence_quote":"The qualitative edge eigenvector universality result whose convergence rate this paper makes explicit."},{"cited_title":"910, viii–100","cited_arxiv_id":null,"evidence_quote":"Gives the edge eigenvalue location $\\lambda_2=2+O(N^{-2/3+o(1)})$ used throughout the rigidity and spacing estimates."},{"cited_title":"Dyson, A brownian-motion model for the eigenvalues of a random matrix , Journal of Mathematical Physics 3 (1962), no","cited_arxiv_id":null,"evidence_quote":"Origin of the Brownian-motion eigenvalue model that the constrained flow generalizes."},{"cited_title":"2, 311–323","cited_arxiv_id":null,"evidence_quote":"Supplies the multivariate Berry-Esseen theorem used to get the convex-set rate in the joint CLT."},{"cited_title":"3B, 2279–2375","cited_arxiv_id":null,"evidence_quote":"Supplies the local semicircle law used in the resolvent correction estimates for the overlap SDE."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sparse-matrix local law and edge rigidity used near the spectral edge."},{"cited_title":"3, 1435–1515","cited_arxiv_id":null,"evidence_quote":"Provides eigenvalue rigidity and eigenvector moment bounds used in the GOE comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the eigenvector distribution results behind the GOE Berry-Esseen bound for overlaps."},{"cited_title":"Haussmann and Etienne Pardoux, Time reversal of diffusions, Annals of Probability 14 (1986), no","cited_arxiv_id":null,"evidence_quote":"Supplies the time-reversal theory for diffusions that underlies the backward stability estimate."}],"review_version":1}