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Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

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Pith's one-line read 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…

desk verdict The edge variance sigma^2=1 and eigenvalue-eigenvector independence look right, but the proof leans on an unpublished companion for a load-bearing moment bound. read the letter →

arxiv 2502.08897 v1 pith:OVV5P54Y submitted 2025-02-13 math.PR math-phmath.COmath.MPmath.SP

Yukun He , Jiaoyang Huang , Horng-Tzer Yau This is my paper · ORCID
classification math.PRmath-phmath.COmath.MPmath.SP MSC 60B2005C8060F05
keywords randomd-regulargraphsedgeeigenvectorsGaussianwavesAiry_1pointprocessuniversalityGreen'sfunctionslocalresamplingregulargraphspectra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $d$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 2.5 and Section 4.1, Eq. (2.30) and Eqs. (4.5)-(4.6)] 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.
  2. [Section 3.2, Step 3, Eqs. (3.40)-(3.44)] 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.
  3. [Section 3.2, Proposition 3.4, Eq. (3.30)] 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.
minor comments (4)
  1. [Proof of Lemma 3.2, Eq. (3.25)] 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).
  2. [Section 3.2, Step 5, around Eq. (3.55)] 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}.
  3. [Theorem 1.1 and Remark 1.3] 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.
  4. [Introduction, Section 1] There are some minor presentation issues, including duplicated reference '[63, 63, 64]' in the first paragraph, which should be cleaned up before publication.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Gaussian-wave eigenvector limit is proved from local resampling and MGF computations rather than assumed; the Airy_1 eigenvalue part is imported from the authors' own earlier work, a directional dependency but not a circular reduction.

full rationale

The paper's new eigenvector claims are derived, not presupposed. Step 3 of Proposition 3.5 computes the conditional MGF of sqrt(N)<v(i),u_s> under the switching randomness and shows asymptotic joint Gaussianity; Step 4 identifies the covariance by an explicit graph sum (3.49) that tends to Cov[Psi(i)Psi(j)] from (1.1). No fitted parameter is renamed as a prediction: the variance 1 emerges from the coefficient A = d(d-1)/(d-2)^2 and the eigenvalue equation, not from a fit. The eigenvalue convergence is explicitly credited to [42]: Remark 1.2 says 'The joint convergence of the extreme eigenvalues to the Airy 1 point process follows from the edge universality result in [42]', and Theorems 2.14-2.15 plus estimate (2.30) from [42, Cor C.3] are used as inputs for rigidity, edge universality, and the tightness in Proposition 3.4. These are genuine prior results of the same authors, but they are not consequences of the present theorem and do not contain the Gaussian-wave or independence conclusions, so the dependency is directional rather than circular. The omitted proof of the Airy bound (3.30), deferred to [70, Proposition 2.4], is likewise an outside input. The main risk is verification risk of the unpublished companion [42], not a logical reduction of the paper's claim to its own assumptions. Score 2 reflects the heavy self-citation load, not a circular derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new parameters fitted to data and no new postulated entities. Its new ingredients are analytic: a Green's function imaginary-part framework and a variance identification. The substantive mathematical inputs are prior results from [23,42,45], mostly by the same authors, which are used as external benchmarks rather than being assumed in a circular way.

assumptions (6)
  • domain assumption Random d-regular graph model and the locally tree-like event Omega (Definition 2.5)
    The graph is sampled uniformly from simple d-regular graphs, and the proofs condition on Omega, which holds with probability 1 - O(N^{-(1-c)omega_d}) by Proposition 2.6.
  • domain assumption Optimal rigidity and edge universality from [42] (Theorems 2.14 and 2.15)
    These results are quoted without proof and are load-bearing for the Airy_1 eigenvalue convergence and for eigenvalue concentration estimates used in the Green's function tail bounds.
  • domain assumption Local resampling exchangeability from [45, Lemma 7.3]
    Lemma 2.4 gives (G, T_S(G)) exchangeable, which is the cornerstone of the Green's function comparison between the original and switched graphs.
  • domain assumption Uniqueness and covariance of Gaussian waves from [23]
    The limiting object and its covariance formula are imported from Elon's construction; the paper only identifies the covariance of the limit through tree Green's functions.
  • standard math Airy_1 point process properties, including the bound (3.30)
    Proposition 3.4 uses the Airy_1 process tightness and almost sure growth bound; the proof is omitted and cited to [70, Proposition 2.4].
  • standard math Skorokhod representation theorem and vague convergence framework
    Used in the proof of Theorem 1.1 to pass to almost sure subsequences after establishing weak convergence of harmonic functions and point processes.

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Cite this review

Pith. "Pith review of Gaussian Waves and Edge Eigenvectors of Random Regular Graphs." pith.science (2026). https://pith.science/paper/OVV5P54Y

@misc{pith2026250208897,
  author       = {Pith},
  title        = {Pith review of: Gaussian Waves and Edge Eigenvectors of Random Regular Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVV5P54Y}},
  note         = {Machine review of arXiv:2502.08897}
}
abstract

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\leq \sigma^2\leq 1$. In this paper, we present an alternative proof of this result for the edge eigenvectors of random regular graphs, establishing that the variance must be $\sigma^2=1$. Furthermore, we show that the eigenvalues and eigenvectors are asymptotically independent. Our approach introduces a simple framework linking the weak convergence of the imaginary part of the Green's function to the convergence of eigenvectors, which may be of independent interest.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants

    math.PR 2025-07 reject novelty 7.0 of 10

    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.

Reference graph

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