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arxiv: 2507.02298 · v3 · submitted 2025-07-03 · 🧮 math.PR

Local laws and spectral properties of deformed sparse random matrices

Ji Oon Lee , Inyoung Yeo This is my paper

Pith reviewed 2026-05-19 06:59 UTC · model grok-4.3

classification 🧮 math.PR
keywords random matriceslocal lawssparse matricesdeformed semicircle laweigenvalue rigidityasymptotic normalityStieltjes transform
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The pith

Deformed sparse random matrices obey local laws that match a refined deformed semicircle law, yielding eigenvalue rigidity and normal edge fluctuations.

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

The paper establishes local laws for the eigenvalues of a sparse symmetric random matrix W deformed by an independent diagonal term λV. These laws establish that the empirical spectral measure stays close to a deterministic refined deformed semicircle law at fine scales. The argument proceeds by controlling the resolvent under mild moment bounds and independence. A reader would care because local spectral control is the standard route to precise statements about eigenvalue locations and fluctuations in high-dimensional models. The same local laws are then used to prove that all eigenvalues are rigid around their classical positions and that the largest and smallest eigenvalues are asymptotically normal after centering and scaling.

Core claim

We consider deformed sparse random matrices of the form H = W + λV, where W is a real symmetric sparse random matrix, V is a random or deterministic real diagonal matrix independent of W, and λ = O(1). Under mild assumptions on the matrix entries of W and V, we prove local laws for H that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of H, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues.

What carries the argument

Local laws for the resolvent of H that provide high-probability entrywise comparison to the deterministic equivalent given by the refined deformed semicircle law.

If this is right

  • All eigenvalues of H concentrate around the quantiles of the deformed semicircle law.
  • The largest and smallest eigenvalues of H, suitably centered and scaled, converge in distribution to a standard normal.
  • The local laws hold with high probability uniformly for spectral parameters away from the edges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same comparison technique could apply when the perturbation V is nondiagonal but still independent of W.
  • The results suggest a route to local laws for sparse matrices arising from weighted graphs with added vertex potentials.
  • Numerical checks near the boundary of the moment assumptions would test how sharp the mild conditions really are.

Load-bearing premise

The entries of the sparse matrix W and the diagonal V satisfy only mild moment bounds and remain independent of each other.

What would settle it

A numerical computation of the local eigenvalue density for a sequence of sparse matrices whose entries have infinite fourth moment, showing systematic deviation from the predicted deformed semicircle law at mesoscopic scales.

read the original abstract

We consider deformed sparse random matrices of the form $H= W+ \lambda V$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $\lambda = O(1) $ is a coupling constant. Under mild assumptions on the matrix entries of $W$ and $V$, we prove local laws for $H$ that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of $H$, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies deformed sparse random matrices of the form H = W + λV, with W a real symmetric sparse random matrix, V a random or deterministic diagonal matrix independent of W, and λ = O(1). Under mild moment and independence assumptions on the entries of W and V, the authors establish local laws comparing the empirical spectral measure of H to a refined deformed semicircle law. These local laws are then applied to prove eigenvalue rigidity and the asymptotic normality of the extremal eigenvalues.

Significance. If the local laws and their consequences hold, the results extend existing techniques for deformed Wigner matrices to the sparse regime, providing a refined comparison to the deformed semicircle law together with rigidity and edge statistics. This is a natural and useful contribution to the study of spectral properties of sparse random matrices, with potential relevance to random graphs and network models. The paper credits the use of Stieltjes-transform estimates and moment methods under explicit assumptions.

major comments (2)
  1. [§2, Theorem 2.1] §2, Theorem 2.1 (main local law): the error term in the comparison to the refined deformed semicircle law is stated to be o(1) with high probability under the mild moment assumptions, but the proof sketch in §4 does not explicitly verify that the sparsity-induced variance fluctuations remain controlled uniformly down to the local scale 1/N; a concrete bound on the variance profile would strengthen the claim.
  2. [§5, Theorem 5.2] §5, Theorem 5.2 (normality of extremal eigenvalues): the reduction from the local law to the CLT for the largest eigenvalues relies on the standard Green-function comparison, yet the paper does not quantify how the sparsity parameter affects the variance of the limiting Gaussian; this step is load-bearing for the edge-statistics claim.
minor comments (2)
  1. [Definition 1.3] Notation for the refined deformed semicircle law is introduced in Definition 1.3 but the explicit formula for the Stieltjes transform m(z) is only derived later in §3; moving the formula forward would improve readability.
  2. [Introduction] Several references to prior works on dense deformed matrices (e.g., Pastur, Shcherbina) are cited in the introduction but the precise technical differences introduced by sparsity are not summarized in a dedicated paragraph.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. We address each major comment below and will incorporate clarifications where appropriate in the revised version.

read point-by-point responses

  1. Referee: [§2, Theorem 2.1] §2, Theorem 2.1 (main local law): the error term in the comparison to the refined deformed semicircle law is stated to be o(1) with high probability under the mild moment assumptions, but the proof sketch in §4 does not explicitly verify that the sparsity-induced variance fluctuations remain controlled uniformly down to the local scale 1/N; a concrete bound on the variance profile would strengthen the claim.

    Authors: We agree that an explicit verification of the variance control would improve the presentation. In the proof of Theorem 2.1 the moment assumptions on the entries of W already ensure that the sparsity-induced fluctuations are of order o(1) uniformly down to scale 1/N, but this is only sketched. We will add a short lemma (or remark) in §4 that supplies a concrete bound on the variance profile, making the uniformity explicit. This is a minor addition that does not alter the statement of the theorem. revision: yes

  2. Referee: [§5, Theorem 5.2] §5, Theorem 5.2 (normality of extremal eigenvalues): the reduction from the local law to the CLT for the largest eigenvalues relies on the standard Green-function comparison, yet the paper does not quantify how the sparsity parameter affects the variance of the limiting Gaussian; this step is load-bearing for the edge-statistics claim.

    Authors: The Green-function comparison argument in §5 yields a limiting Gaussian whose variance is determined by the second-moment structure of the entries of W. Under the normalization and independence assumptions of the paper, this variance is independent of the sparsity level (it coincides with the dense deformed Wigner case). Nevertheless, we acknowledge that an explicit formula would be helpful. We will insert a short paragraph after the statement of Theorem 5.2 that records the precise expression for the limiting variance in terms of the model parameters, thereby quantifying any dependence on sparsity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external techniques

full rationale

The paper derives local laws for the resolvent of H = W + λV by establishing Stieltjes-transform estimates that compare the empirical spectral measure to a refined deformed semicircle law, under explicit moment bounds and independence of W and V. These estimates close directly from the assumptions without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. Eigenvalue rigidity and extremal normality then follow from the local laws by standard arguments in random matrix theory. The chain is independent of the target results and relies on external methods, so the derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard moment assumptions and independence but introduces no new free parameters or invented entities visible in the abstract; the refined deformed semicircle law is defined via the model parameters.

axioms (1)
  • domain assumption Entries of W satisfy mild moment bounds and V entries are independent of W
    Invoked to control the resolvent and close the local law estimates

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Ajanki, O., Erd˝ os, L., Kr¨ uger, T.:Quadratic Vector Equations on Complex Upper Half-Plane , Mem. Am. Math. Soc. 261 (2019)

    work page 2019

  2. [2]

    Alt, J., Ducatez, R., Knowles, A.: Delocalization Transition for Critical Erd˝ os-R´ enyi Graphs, Commun. Math. Phys. 388, 507-579 (2021)

    work page 2021

  3. [3]

    Alt, J., Ducatez, R., Knowles, A.: Localized Phase for the Erd˝ os-R´ enyi Graphs, Commun. Math. Phys. 405, 9 (2024)

    work page 2024

  4. [4]

    Alt, J., Erd˝ os, L., Kr¨ uger, T.:The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps , Doc. Math. 25, 1421-1539 (2020)

    work page 2020

  5. [5]

    Alt, J., Erd˝ os, L., Kr¨ uger, T., Schr¨ oder, D.:Correlated Random Matrices: Band Rigidity and Edge Universality , Ann. Probab. 48(2), 963-1001 (2020)

    work page 2020

  6. [6]

    Bao, Z., Erd˝ os, L., Schnelli, K.: Local Law of Addition of Random Matrices on Optimal Scale , Commun. Math. Phys. 349, 947-990 (2017)

    work page 2017

  7. [7]

    Bao, Z., Erd˝ os, L., Schnelli, K.: On the Support of the Free Additive Convolution , J. Anal. Math. 142, 323-348 (2020)

    work page 2020

  8. [8]

    Bao, Z., Erd˝ os, L., Schnelli, K.: Spectral Rigidity for Addition of Random Matrices at the Regular Edge , J. Funct. Anal. 279, 108639 (2020)

    work page 2020

  9. [9]

    Bauerschmidt, R., Huang, J., Knowles, A., Yau, H.-T.: Edge Rigidity and Universality of Random Regular Graphs of Intermediate Degree, Geom. Funct. Anal. 30, 693–769 (2020)

    work page 2020

  10. [10]

    Belinschi, S., Bercovici, H.: A New Approach to Subordination Results in Free Probability , J. Anal. Math. 101, 357-365 (2007)

    work page 2007

  11. [11]

    Benigni., L.: Eigenvectors Distribution and Quantum Unique Ergodicity for Deformed Wigner Matrices , Ann. Inst. H. Poincar´ e Probab. Statist.56(4), 2822-2867 (2020)

    work page 2020

  12. [12]

    Biane, P.: On the Free Convolution with a Semi-circular Distribution , Indiana Univ. Math. J. 46, 705-518 (1997)

    work page 1997

  13. [13]

    Blomendal, A., Erd˝ os, L., Knowles, A., Yau, H.-T., Yin, J.: Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices, Electron. J. Probab. 18(59), (2013)

    work page 2013

  14. [14]

    C.: On the spectral edge of non-Hermitian random matrices , arXiv:2404.17512 (2024)

    Campbell, A., Cipolloni, G., Erd˝ os, L., Ji, H. C.: On the spectral edge of non-Hermitian random matrices , arXiv:2404.17512 (2024)

    work page arXiv 2024

  15. [15]

    Sigma 11, 74, (2023)

    Cipolloni, G., Erd˝ os, L., Hanheik, J., Kolupaiev, O.: Gaussian Fluctuations in the Equipartition Principle for Wigner Matrices, Forum Math. Sigma 11, 74, (2023)

    work page 2023

  16. [16]

    Erd˝ os, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erd˝ os-R´ enyi Graphs I: Local Semicircle Law, Ann. Probab. 41, 2279-2375 (2013)

    work page 2013

  17. [17]

    Erd˝ os, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erd˝ os-R´ enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues , Commun. Math. Phys. 314(3), 587-640 (2012)

    work page 2012

  18. [18]

    Sigma 7, 8, (2023)

    Erd˝ os, L., Kr¨ uger, T., Schr¨ oder, D.:Random Matrices with Slow Correlation Decay , Forum Math. Sigma 7, 8, (2023). 57

    work page 2023

  19. [19]

    Erd˝ os, L., Yau, H.-T.:A Dynamical Approach to Random Matrix Theory , Courant Lecture Notes in Mathematics 28, Courant Institute of Mathematical Sciences, New York, NY; American Mathematical Society, Providence, RI, (2017)

    work page 2017

  20. [20]

    O., Yang, W.: Spectral Properties and Weak Detection in Stochastic Block Models , Random Matrices Theory Appl

    Han, Y., Lee, J. O., Yang, W.: Spectral Properties and Weak Detection in Stochastic Block Models , Random Matrices Theory Appl. 14(2), 2550003 (2025)

    work page 2025

  21. [21]

    He, Y.: Bulk Eigenvalue Fluctuations of Sparse Random Matrices , Ann. Appl. Probab. 30(6), 2846–2879 (2020)

    work page 2020

  22. [22]

    He, Y.: Edge Universality of sparse Erd˝ os-R´ enyi Digraphs, arXiv:2304.04723 (2023)

    work page arXiv 2023

  23. [23]

    He, Y.: Spectral Gap and Edge Universality of Dense Random Regular Graph , Commun. Math. Phys. 405, 181 (2024)

    work page 2024

  24. [24]

    He, Y., Huang, J., Wang, C.: Extremal eigenvectors of sparse random matrices , arXiv:2501.16444 (2025)

    work page arXiv 2025

  25. [25]

    Theory Rel

    He, Y., Knowles, A.: Fluctuations of Extreme Eigenvalues of Sparse Erd˝ os-R´ enyi Graphs, Probab. Theory Rel. Fields 180, 985-1056 (2021)

    work page 2021

  26. [26]

    Theory Rel

    He, Y., Knowles, A., Rosenthal, R.: Isotropic Self–consistent Equations for Mean–field Random Matrices , Probab. Theory Rel. Fields 171, 203-249 (2018)

    work page 2018

  27. [27]

    Huang, J., Landon, B.: Spectral Statistics of Sparse Erd˝ os-R´ enyi Graph Laplacians, Ann. Inst. H. Poincar´ e Probab. Statist. 56(1), 120-154 (2020)

    work page 2020

  28. [28]

    Huang, J., Landon, B., Yau, H.-T.: Bulk Universality of Sparse Random Matrices , J. Math. Phys. 56(12), 123301 (2015)

    work page 2015

  29. [29]

    Huang, J., Landon, B., Yau, H.-T.: Transition from Tracy–Widom to Gaussian Fluctuations of Extremal Eigen- values of Sparse Erd˝ os-R´ enyi Graphs, Ann. Probab. 48, 239-298 (2020)

    work page 2020

  30. [30]

    Huang, J., Yau, H.-T.: Edge Universality of Sparse Random Matrices , arXiv:2206.06580 (2022)

    work page arXiv 2022

  31. [31]

    Pure Appl

    Huang, J., Yau, H.-T.: Spectrum of Random d-regular Graphs up to the Edge , Commun. Pure Appl. Math. 77(3), 1635–1723 (2024)

    work page 2024

  32. [32]

    C., Lee, J

    Ji, H. C., Lee, J. O.: Gaussian fluctuations for linear spectral statistics of deformed Wigner Matrices , Random Matrices Theory Appl. 9(3), 2050011 (2020)

    work page 2020

  33. [33]

    C., Park, J.: Tracy–Widom limit for Free Sum of Random Matrices , Ann

    Ji, H. C., Park, J.: Tracy–Widom limit for Free Sum of Random Matrices , Ann. Probab. 53, 239-298 (2025)

    work page 2025

  34. [34]

    Kargin, V., Subordination for the Sum of Two Random Matrices , Ann. Probab. 43(4), 2119-2150 (2015)

    work page 2015

  35. [35]

    Theory Rel

    Knowles, A, Yin, J.: Anisotropic Local Laws for Random Matrices, Probab. Theory Rel. Fields 169, 257-352 (2017)

    work page 2017

  36. [36]

    Lee, J.: Higher Order Fluctuations of Extremal Eigenvalues of Sparse Random Matrices , Ann. Inst. H. Poincar´ e Probab. Statist. 60(4), 2694-2735 (2024)

    work page 2024

  37. [37]

    O., Li, Y.: Spherical Sherrington-Kirkpatrick Model for Deformed Wigner Matrix with Fast Decaying Edges, J

    Lee, J. O., Li, Y.: Spherical Sherrington-Kirkpatrick Model for Deformed Wigner Matrix with Fast Decaying Edges, J. Stat. Phys. 190, 35 (2023)

    work page 2023

  38. [38]

    O., Schnelli, K.: Local Deformed Semicircle Law and Complete Delocalization for Wigner Matrices with Random Potential, J

    Lee, J. O., Schnelli, K.: Local Deformed Semicircle Law and Complete Delocalization for Wigner Matrices with Random Potential, J. Math. Phys. 54 103504 (2013)

    work page 2013

  39. [39]

    O., Schnelli, K.: Edge Universality for Deformed Wigner Matrices, Rev

    Lee, J. O., Schnelli, K.: Edge Universality for Deformed Wigner Matrices, Rev. Math. Phys. 27(8), 1550018 (2015)

    work page 2015

  40. [40]

    O., Schnelli, K.: Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices , Probab

    Lee, J. O., Schnelli, K.: Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices , Probab. Theory Relat. Fields 164, 165-241 (2016)

    work page 2016

  41. [41]

    O., Schnelli, K.: Local law and Tracy–Widom limit for Sparse Random Matrices, Probab

    Lee, J. O., Schnelli, K.: Local law and Tracy–Widom limit for Sparse Random Matrices, Probab. Theory Rel. Fields 171, 543-616 (2018)

    work page 2018

  42. [42]

    O., Schnelli, K., Stetler, B., Yau, H.T.: Bulk Universality for Deformed Wigner Matrices , Ann

    Lee, J. O., Schnelli, K., Stetler, B., Yau, H.T.: Bulk Universality for Deformed Wigner Matrices , Ann. Probab. 44(3), 2349-2425 (2016)

    work page 2016

  43. [43]

    Li, Y., Schnelli, K., Xu, Y.: Central Limit Theorem for Mesoscopic Eigenvalue Statistics of Deformed Wigner Matrices and Sample Covariance Matrices , Ann. Inst. H. Poincar´ e Probab. Statist.57(1), 506-546 (2021)

    work page 2021

  44. [44]

    3(2), 1450005 (2014)

    O’Rourke, S., Vu, V.: Universality of Local Eigenvalue Statistics in Random Matrices with External Source, Random Matrices Theory Appl. 3(2), 1450005 (2014)

    work page 2014

  45. [45]

    , Theoret

    Pastur, L.: On the spectrum of random matrices. , Theoret. Math. Phys. 10(1), 67–74, 1972

    work page 1972

  46. [46]

    Pastur, L., Vasilchuk, V.: On the Law of Addition of Random Matrices, Commun. Math. Phys. 214, 249-286 (2000)

    work page 2000

  47. [47]

    Shcherbina, M.: On universality of local edge regime for the deformed Gaussian unitary ensemble , J. Stat. Phys. 143, 455-481 (2011)

    work page 2011

  48. [48]

    Shcherbina, M., Tirozzi, B.: Central Limit Theorem for Fluctuations of Linear Eigenvalue Statistics of Large Random Graphs: Diluted Regime , J. Math. Phys. 53, 043501 (2012)

    work page 2012

  49. [49]

    Stone, B., Yang, F., Yin, J.: A Random Matrix Model Towards the Quantum Chaos Transition Conjecture , Com- mun. Math. Phys. 406, 85, (2025)

    work page 2025

  50. [50]

    V., Dykema, K

    Voiculescu, D. V., Dykema, K. J., Nica, A.: Free Random Variables: A Noncommutative Probability Approach to Free Products With Applications to Random Matrices, Operator Algebras, and Harmonic Analysis on Free Groups, CRM Monograph Series 1 American Mathematical Society, Providence, RI, (1992)

    work page 1992

  51. [51]

    von Soosten, P., Warzel, S.: Non-ergodic Delocalization in the Rosenzweig–Porter Model , Lett. Math. Phys. 109, 905–922 (2019)

    work page 2019