{"id":"ec263e92-0f52-4c94-9b58-9d7a8ef37d41","arxiv_id":"2502.08897","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Edge eigenvectors of random d-regular graphs converge to Gaussian waves with variance 1, jointly with and asymptotically independent of the Airy_1 edge eigenvalue process.","lead":"For random d-regular graphs, the paper proves that the top edge eigenvectors, viewed on any fixed ball, converge to the standard Gaussian wave process on the infinite d-regular tree, with variance exactly one, and that these eigenvectors are asymptotically independent of the Airy_1 edge eigenvalues. This pins down the edge case of the Backhausz-Szegedy Gaussian wave theorem and provides a graph-theoretic analogue of Berry's random wave conjecture.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The joint eigenvalue-eigenvector convergence rests on the optimal-scale moment bound (4.5), which is derived solely from the companion estimate (2.30) of [42]; an error in that unverified loop-equation corollary would invalidate the tightness (Prop. 3.4) and hence the truncation at (3.55).","rationale":"The paper presents a detailed proof of a significant result, and the internal arguments are largely checkable. The most vulnerable point is exactly the one identified by the reader: the proof of the optimal-scale local law at the edge (Proposition 4.3) depends on the moment estimate (2.30) from the companion unpublished preprint [42]. This estimate is load-bearing because it underpins the tightness results in Proposition 3.4, which are needed in Step 1 to obtain joint convergence of eigenvalues and Stieltjes transform, and in Step 5 to truncate the Green's function expansion. If (2.30) has an error or an unstated condition, the central claims of eigenvalue-eigenvector independence and variance σ²=1 lose their proof. I did not find a fatal internal inconsistency: the conditioning on the event I(F,G)=1 in the switching MGF computation (Section 3.2, Step 3) is not written rigorously, but the event has probability 1-O(N^{-1+2c}) and the integrand is subexponential on the bad event, so this gap appears repairable. The omitted proof of (3.30) for the Airy point process is also a minor self-containment issue. Therefore the verdict should remain CONDITIONAL, with the condition being a positive verification of the companion estimate (2.30).","tokens_in":41738,"tokens_out":24143,"duration_ms":228246,"concrete_test":"Independently re-derive [42, Corollary C.3] (the moment bound (2.30)) from the loop equations of the companion paper, carefully checking the exponent of (d-1)^{-ℓ/4} in the last term and the definitions of Φ, Υ, and ~Υ. If (2.30) does not imply E[1(G∈Ω)|Q-msc|^{2p}] ≲ (Nη)^{-2p} for Nη√(κ+η) ≥ C, then Proposition 4.3 is false as stated and the tightness of the Stieltjes transform in Proposition 3.4 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is proved through Proposition 3.5, whose Step 5 controls the tail of the Green's function expansion using the tightness of the rescaled eigenvalue counting measure YN (from Proposition 3.4) and the Gaussian concentration of the switching variables. Proposition 3.4 is proved from Propositions 4.1 and 4.2, which in turn are consequences of Proposition 4.3. Proposition 4.3's key estimates (4.5) and (4.6) are obtained in Section 4.1 by feeding the moment bound (2.30), quoted as [42, Corollary C.3], into the quadratic equation (4.14) and the Stieltjes-transform expansion (4.12). If (2.30) is wrong — for instance, if the exponent of (d-1)^{-ℓ/4} in the last term is incorrect, or if the bound requires an extra spectral condition near the edge — then the conclusion E[1(G∈Ω)|Q-msc|^{2p}] ≲ (Nη)^{-2p} in (4.26) does not follow at the optimal scale Nη√(κ+η) ≥ C. Consequently the tightness of the Stieltjes transform (3.32) and of YN in Proposition 3.4 fails, Step 1 of Proposition 3.5 cannot justify the joint convergence of eigenvalues and the Stieltjes transform, and the truncation in (3.55) is not controlled. This directly threatens the eigenvalue-eigenvector independence and the variance σ²=1 claims. The estimate (2.30) is a corollary of loop equations in an unpublished companion preprint [42] (arXiv:2412.20263), and its validity is not checked in the present paper. The proof of the Airy-process bound (3.30) is also omitted with a citation to [70], a smaller but additional gap in Proposition 3.4.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for fixed d, the top O(1) edge eigenvalues of a random d-regular graph, rescaled by (AN)^{2/3}, and the corresponding eigenvectors restricted to any fixed-radius ball, jointly converge to the Airy_1 point process and to independent copies of the Gaussian wave on the infinite d-regular tree, with covariance (1.1) and hence variance sigma^2=1. The proof introduces a Green's-function framework: local resampling of boundary edges expresses the edge Green's function as a quadratic form in randomized boundary data; a Gaussian moment-generating-function computation identifies the limiting covariance; and a tightness/truncation argument controls the tail. The main theorem is the first proof of sigma^2=1 for edge eigenvectors as well as asymptotic independence of edge eigenvalues and eigenvectors.","tokens_in":42123,"tokens_out":11689,"duration_ms":117260,"significance":"If the result is correct, it resolves the variance ambiguity left open by Backhausz and Szegedy for edge eigenvectors, establishes sigma^2=1, and gives asymptotic independence between edge eigenvalues and eigenvectors. The Green's-function-to-eigenvector framework in Section 1.3 may be reusable for other sparse random matrix models. Strengths include a precise, falsifiable statement, no fitted parameters, detailed error bounds in Sections 3.1 and 4.1, and an explicit covariance formula. The main caveat is that several central inputs, in particular the optimal rigidity and edge universality of [42] and the moment estimate (2.30), are outsourced to an unpublished companion paper, and a tail bound for the Airy_1 process is cited to another preprint.","major_comments":[{"comment":"The tightness of the rescaled eigenvalue counting measure and of the Stieltjes transform (Proposition 3.4), and through them the truncation at Eq. (3.55) in Step 5 of Proposition 3.5, is ultimately based on the moment estimate quoted as [42, Corollary C.3] in Eq. (2.30). This estimate is fed into the quadratic equation (4.14) and the expansion (4.12) to obtain Proposition 4.3. The companion paper [42] is an unpublished preprint, and the estimate is neither reproduced nor proved here. If (2.30) were incorrect or required an additional condition near the edge, then the conclusions (4.5)-(4.6), the tightness of Y_N, and the decomposition (3.55) would not be justified. The authors should either prove (2.30) in this paper or include a complete, self-contained statement with a verifiable proof; otherwise the central claim remains conditional on an unverified black box.","section":"Section 2.5 and Section 4.1, Eq. (2.30) and Eqs. (4.5)-(4.6)"},{"comment":"The Gaussian approximation of \\sqrt{N}\\langle v^{(i)},u_s\\rangle uses the moment-generating-function factorization E_S[\\prod_\\alpha \\exp(\\cdots)] = \\prod_\\alpha E_S[\\cdots], which requires independence of X_s(\\alpha) over \\alpha. Section 2.2, however, explicitly allows repetitions in the resampling data (b_\\alpha,c_\\alpha). With positive probability the same oriented edge is chosen twice, and in that case the displayed factorization is not literal. The paper should either forbid repetitions in the admissible data or condition on a no-collision event and estimate the probability and error introduced by that conditioning. This is a local but real gap in the proof of the Gaussian convergence of the random boundary sums.","section":"Section 3.2, Step 3, Eqs. (3.40)-(3.44)"},{"comment":"Proposition 3.4 states that Y = \\sup_{x\\le 0}(1+|x|)^{-2/3}|\\{i: A_i \\ge -x\\}| < \\infty almost surely for the Airy_1 point process and says 'We omit the proof', citing [70, Proposition 2.4], which is another unpublished preprint. This bound is used to justify the absolute convergence of the limiting series (3.34) and the limiting tail estimate (3.57). A proof should be included, or at minimum the precise statement of the cited result should be reproduced so the validity of the bound can be checked independently.","section":"Section 3.2, Proposition 3.4, Eq. (3.30)"}],"minor_comments":[{"comment":"In the bound for Im[III], the final estimate is written as \\lesssim N^{-1/3+b/2}; the sign of the exponent appears to be wrong and should be N^{-1/3-b/2}, consistent with (3.21) and (3.23).","section":"Proof of Lemma 3.2, Eq. (3.25)"},{"comment":"The parameter j is introduced only as 'large'. To make the passage from (3.55) to (3.57) rigorous, j should be fixed before sending N to infinity and only afterwards sent to infinity, since the Gaussian approximation in Step 3 is proved for 2 \\le s \\le \\sqrt{N}.","section":"Section 3.2, Step 5, around Eq. (3.55)"},{"comment":"The proof of Theorem 1.1 establishes convergence of the rank-one products N u_s(i)u_s(j), while the theorem is formulated with random-sign eigenvectors. Remark 1.3 asserts the equivalence but does not prove it; a short justification would remove ambiguity, for instance by using conditional symmetry of the signs and the fact that the limiting Gaussian variables are nonzero almost surely.","section":"Theorem 1.1 and Remark 1.3"},{"comment":"There are some minor presentation issues, including duplicated reference '[63, 63, 64]' in the first paragraph, which should be cleaned up before publication.","section":"Introduction, Section 1"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved issue is the dependence on [42], an unpublished preprint co-authored by two of the present authors. The key estimate (2.30) and the rigidity/universality inputs are not independently checkable from the material in this paper. I recommend that the editor require either an appendix proving (2.30) and the cited rigidity/universality inputs, or a verifiable version of the companion paper, before acceptance. Apart from this, the paper is a strong contribution with a clear and potentially reusable framework, and it fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious, well-executed paper, and the main new claims—that edge eigenvectors of random d-regular graphs converge to Gaussian waves with variance exactly 1 and are asymptotically independent of the Airy_1 edge eigenvalues—are very plausible. The proof is not self-contained: the tightness of the eigenvalue counting measure and the optimal-scale Stieltjes estimates rest on a moment bound quoted from the companion preprint [42, Cor. C.3], and if that bound fails, the argument collapses. But I did not find a problem in the paper's own sections; the dependency is directional, not circular.\n\nWhat is new: Backhausz–Szegedy had variance in [0,1]; this paper pins it to 1 at the edge and proves independence. The Green's function framework in Section 3.1 is neat and likely reusable; it gives a clean way to get joint eigenvalue-eigenvector convergence from local resampling. Sections 3.1–3.2 are detailed and internally consistent. The covariance computation in Step 4 correctly identifies the Gaussian wave covariance. The random sign issue is handled properly, and the quadratic-products formulation in Remark 1.3 avoids the sign ambiguity. I checked the Schur complement expansions and they are careful.\n\nSoft spots, in proportion. First, the key moment estimate (2.30) underlying Proposition 4.3 is imported from an unpublished companion; the stress-test note is right that (4.5) at Nη√(κ+η) ≥ C is exactly what powers the truncation in Step 5 and the tightness of Y_N. This is a load-bearing external input, not a small detail. Second, the proof of (3.30)—the Airy point process tail bound—is omitted with a citation to [70]; this is minor since a limiting version follows from the paper's own (4.3)–(4.4), but it is still a gap. Third, the paper leans on the authors' previous results for edge universality and optimal rigidity; that is standard practice, and those results are part of the public posting pipeline.\n\nVerdict: conditional accept in spirit. If the companion's Cor. C.3 holds, the result goes through. The paper deserves a serious referee, ideally one who can verify (2.30) and the stated spectral conditions in [42]. Recommend peer review; the authors should be asked to either include the optimal-scale proof of (4.5) or state precisely which conditions from [42] are assumed.","headline":"The edge variance sigma^2=1 and eigenvalue-eigenvector independence look right, but the proof leans on an unpublished companion for a load-bearing moment bound.","tokens_in":42655,"tokens_out":2368,"would_cite":true,"duration_ms":23293,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","05C80","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the extreme eigenvalues and edge eigenvectors of random d-regular graphs converge jointly to the Airy$_1$ point process and independent Gaussian waves, with the two asymptotically independent and with variance…","keywords":["random d-regular graphs","edge eigenvectors","Gaussian waves","Airy_1 point process","edge universality","Green's functions","local resampling","random regular graph spectra"],"falsifier":"Numerically simulate random $d$-regular graphs for fixed $d$, say $d=3$ and $N=10^5$ or larger: take an eigenvector $u_s$ with $(AN)^{2/3}(\\lambda_s-2)$ in a bounded window near the edge, multiply by a random sign, and estimate the empirical covariance of $\\sqrt{N}u_s$ over vertices at tree-distance $r$ within a small ball, averaged over many graphs. If the theorem is right, these covariances tend to $(d-1)^{-r/2}(1+(d-2)r/d)$, with asymptotic Gaussianity of finite collections; a variance limit strictly less than 1, non-Gaussian marginals, or dependence of the covariance on the window width would refute the claim.","tokens_in":41529,"feed_emoji":"🌊","tokens_out":8083,"duration_ms":77498,"temperature":0.7,"pith_summary":"This paper proves that the extreme eigenvalues and edge eigenvectors of random d-regular graphs converge jointly to known universal objects: the Airy$_1$ point process for the eigenvalues and independent Gaussian waves for the eigenvectors, with the two asymptotically independent. The central new claim is that the variance of the limiting Gaussian wave is exactly 1, not merely somewhere in the range $0\\le \\sigma^2\\le 1$ left open by earlier work. If true, this gives a graph version of Berry's random-wave conjecture at the spectral edge and fixes the fluctuation scale of localized eigenvector observables. The proof develops a direct route from weak convergence of the imaginary part of the Green's function to eigenvector convergence, avoiding comparison to Gaussian ensembles.","feed_headline":"Edge eigenvectors converge to Gaussian waves with variance exactly one","feed_subtitle":"Eigenvalues and eigenvectors at the spectral edge converge jointly to the Airy-1 process and independent Gaussian waves, pinning the…","key_machinery":"The carrying object is the Gaussian wave $\\Psi$, the unique Gaussian eigenvector process on the infinite $d$-regular tree with eigenvalue $2\\sqrt{d-1}$ and covariance $(d-1)^{-r/2}(1+(d-2)r/d)$. The argument's mechanism is local resampling: randomize the boundary edges of a radius-$\\ell$ ball around a fixed vertex, making the original and switched graphs an exchangeable pair, then express the imaginary part of the switched Green's function near the edge as a Poisson-kernel integral over randomized boundary data. The identities $\\operatorname{Im}[A^{-1}]=-A^{-1}\\operatorname{Im}[A]\\overline{A}^{-1}$, the Schur complement formula, and the Ward identity turn this into a sum whose terms have Gaussian limits, and a harmonic-function convergence lemma converts Green's function convergence into vague convergence of the associated eigenvector measures.","core_discovery":"The paper's main theorem fixes $d\\ge 3$ and states that for the normalized adjacency matrix $H=A/\\sqrt{d-1}$ of a uniformly random $d$-regular graph, the rescaled second-through-$(k+1)$-th eigenvalues $(AN)^{2/3}(\\lambda_s-2)$ converge jointly to the Airy$_1$ point process, while the corresponding random-signed rescaled eigenvectors $\\sqrt{N}u_s$, restricted to any fixed radius-$r$ ball $B_r(o;G)$, converge jointly to independent copies of the Gaussian wave $\\Psi$ with covariance $\\operatorname{Cov}[\\Psi(i)\\Psi(j)]=(d-1)^{-r/2}(1+(d-2)r/d)$ for $r=\\operatorname{dist}(i,j)$. The eigenvalues and eigenvectors are asymptotically independent. The same statement holds at the bottom edge. In particular the variance of the Gaussian wave is $\\sigma^2=1$, resolving the range $0\\le \\sigma^2\\le 1$ found by earlier work.","pith_inferences":["This Green's-function route should transfer to other locally tree-like sparse random graph models where edge universality holds, such as sparse Erdős–Rényi graphs, yielding the same Gaussian-wave limit.","One would expect edge local eigenvector observables such as nodal counts and quantum-ergodicity sums to show Gaussian fluctuations at the same variance-one scale, an extension the paper raises but does not develop.","A moderate-$N$ simulation could test the covariance profile directly: even before the Airy$_1$ limit sets in, the ratio of empirical covariance to the formula should approach a $d$-dependent constant.","The asymptotic independence of eigenvalues and eigenvectors is stronger than either marginal law, implying that conditioning on the Airy$_1$ spacings does not alter the local Gaussian wave law."],"forward_implications":["The variance of the limiting Gaussian wave is exactly 1, closing the interval left open by earlier almost-eigenvector results.","The rescaled edge eigenvalues and edge eigenvectors converge jointly to the Airy$_1$ point process and independent Gaussian waves, so the two statistics are asymptotically independent.","The same joint convergence holds for the smallest eigenvalues and their associated eigenvectors.","The explicit covariance formula gives a computable correlation profile on radius-$r$ balls that depends only on tree distance and the degree $d$."],"supporting_citations":[{"why":"Establishes that almost eigenvectors of random regular graphs converge to Gaussian waves with variance in $[0,1]$; the present paper supplies an alternative proof and the variance-one conclusion.","marker":"[9]"},{"why":"Companion preprint providing optimal rigidity and edge universality, used as black boxes; also the source of the moment estimate (2.30) behind the Airy$_1$ eigenvalue convergence and the truncation step.","marker":"[42]"},{"why":"Local Kesten–McKay law and local resampling machinery; supplies exchangeability of the switched pair, Green's function extension estimates, and the bounds used throughout the proof.","marker":"[45]"},{"why":"Defines Gaussian waves on the regular tree and gives their uniqueness and covariance structure, which the main theorem identifies as the limiting object.","marker":"[23]"},{"why":"Wigner-matrix eigenvector distribution result whose Green's-function and Poisson-kernel strategy the proof adapts to random regular graphs.","marker":"[48]"},{"why":"Optimal local law for beta-ensembles via loop equations; its method is followed in the tightness section for the Stieltjes transform.","marker":"[17]"},{"why":"Introduces the Airy$_1$ point process and Tracy–Widom distribution for GOE that the edge eigenvalues converge to.","marker":"[68]"}],"fun_headline_variants":["Edge eigenvectors of random regular graphs: Gaussian waves with variance 1","Variance pinned to 1 for Gaussian wave edge eigenvectors of random regular graphs","Random regular graph edge eigenvectors get Gaussian waves with variance exactly 1","Gaussian waves at unit variance for random regular graph edge eigenvectors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument relies on earlier companion results, not reproved here, about where the extreme eigenvalues sit and about their universal edge statistics; those results carry the eigenvalue convergence and an essential tail bound, so if they have an unstated condition or error, the theorem's eigenvalue part and the truncation step fail.","fun_headline_variants_meta":{"raw":{"variants":["Edge eigenvectors of random regular graphs: Gaussian waves with variance 1","Variance pinned to 1 for Gaussian wave edge eigenvectors of random regular graphs","Random regular graph edge eigenvectors get Gaussian waves with variance exactly 1","Gaussian waves at unit variance for random regular graph edge eigenvectors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00161,"raw_usage":{"total_tokens":6363,"prompt_tokens":850,"completion_tokens":5513,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":5437}},"tokens_in":466,"tokens_out":5513,"duration_ms":35689,"temperature":1.0,"reasoning_tokens":5437,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:17:55.002862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically simulate random $d$-regular graphs for fixed $d$, say $d=3$ and $N=10^5$ or larger: take an eigenvector $u_s$ with $(AN)^{2/3}(\\lambda_s-2)$ in a bounded window near the edge, multiply by a random sign, and estimate the empirical covariance of $\\sqrt{N}u_s$ over vertices at tree-distance $r$ within a small ball, averaged over many graphs. If the theorem is right, these covariances tend to $(d-1)^{-r/2}(1+(d-2)r/d)$, with asymptotic Gaussianity of finite collections; a variance limit strictly less than 1, non-Gaussian marginals, or dependence of the covariance on the window width would refute the claim.","supporting_citations":[{"cited_title":"Backhausz and B","cited_arxiv_id":null,"evidence_quote":"Establishes that almost eigenvectors of random regular graphs converge to Gaussian waves with variance in $[0,1]$; the present paper supplies an alternative proof and the variance-one conclusion."},{"cited_title":"Huang and H.-T","cited_arxiv_id":null,"evidence_quote":"Local Kesten–McKay law and local resampling machinery; supplies exchangeability of the switched pair, Green's function extension estimates, and the bounds used throughout the proof."},{"cited_title":"Gaussian Waves on the Regular Tree","cited_arxiv_id":"0907.5065","evidence_quote":"Defines Gaussian waves on the regular tree and gives their uniqueness and covariance structure, which the main theorem identifies as the limiting object."},{"cited_title":"Knowles and J","cited_arxiv_id":null,"evidence_quote":"Wigner-matrix eigenvector distribution result whose Green's-function and Poisson-kernel strategy the proof adapts to random regular graphs."},{"cited_title":"Bourgade, K","cited_arxiv_id":null,"evidence_quote":"Optimal local law for beta-ensembles via loop equations; its method is followed in the tightness section for the Stieltjes transform."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Airy$_1$ point process and Tracy–Widom distribution for GOE that the edge eigenvalues converge to."}],"review_version":1}