pith. sign in

arxiv: 2607.02446 · v1 · pith:OCWY4C7Inew · submitted 2026-07-02 · 🧮 math-ph · cond-mat.dis-nn· cond-mat.stat-mech· math.MP· math.PR

On a Rosenzweig-Porter-type model

Giorgio Cipolloni , L\'aszl\'o Erd\H{o}s , Joscha Henheik This is my paper

Pith reviewed 2026-07-03 03:34 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.dis-nncond-mat.stat-mechmath.MPmath.PR
keywords Rosenzweig-Porter modeldeformed Wigner matriceslocal lawseigenvector localizationeigenstate thermalization hypothesismobility edgere-entrant localization
0
0 comments X

The pith

Local laws for the resolvent of H = H0 + λW hold uniformly in arbitrary Hermitian H0 and all λ.

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

The paper studies the Rosenzweig-Porter model H = H0 + λW with arbitrary Hermitian H0 and Wigner W. It proves that local laws on the resolvent hold uniformly across all choices of H0 and all values of the coupling λ. This uniformity makes it possible to follow how eigenvectors change from localized at λ = 0 to delocalized at large λ and to locate where the eigenstate thermalization hypothesis begins to hold. The argument rests on controlling a deterministic approximation to the resolvent whose structure changes strongly with the spectrum of H0. The same control yields a mobility edge and re-entrant localization as by-products.

Core claim

We consider a very general Rosenzweig-Porter-type model, H = H0 + λW, where H0 is an arbitrary Hermitian matrix and W is a standard Wigner matrix. We precisely trace the localization properties of the eigenvectors and the eigenstate thermalisation hypothesis (ETH) as the coupling constant λ interpolates between the trivial λ = 0 case and the fully mean field regime of large λ. Our results hold uniformly in H0 and λ, substantially generalising all previous local laws on deformed Wigner matrices even in the mean field regime. Our proof precisely captures the deterministic approximation to the resolvent which exhibits a strongly inhomogeneous structure. As a byproduct, we conclude the emergence

What carries the argument

The deterministic approximation to the resolvent, which has a strongly inhomogeneous structure determined by the spectrum of H0.

If this is right

  • Eigenvector localization lengths vary continuously with λ and can be read off from the local laws.
  • The eigenstate thermalization hypothesis holds once λ exceeds a threshold determined by the spectrum of H0.
  • A mobility edge appears in the spectrum separating localized and extended eigenvectors.
  • Re-entrant localization occurs for certain ranges of λ and eigenvalue locations.

Where Pith is reading between the lines

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

  • The uniformity in H0 suggests the same approximation can be used for any fixed base Hamiltonian without re-deriving constants.
  • The inhomogeneous resolvent structure may appear in other models where a deterministic part is added to a random matrix.
  • The mobility-edge result supplies a concrete prediction for the transition point that could be checked in finite-N simulations of structured perturbations.

Load-bearing premise

The deterministic approximation to the resolvent exhibits a strongly inhomogeneous structure that the proof can capture precisely.

What would settle it

A direct numerical computation of the resolvent entries for a fixed diagonal H0 at an intermediate λ that deviates from the predicted inhomogeneous deterministic approximation by more than the claimed error bound.

Figures

Figures reproduced from arXiv: 2607.02446 by Giorgio Cipolloni, Joscha Henheik, L\'aszl\'o Erd\H{o}s.

Figure 1
Figure 1. Figure 1: Typical shape of the profile i 7→ (Im Mλ)ii for a diagonal deterministic H0, using the notation ν := η + λ 2ρ. In particular, (Im Mλ)ii is very different from the average ⟨Im Mλ⟩ as customary in mean-field models. Since understanding the resolvent G and its deterministic approximation M is a starting point of most current techniques in RMT, the structure of M has a profound effect on the entire theory. The… view at source ↗
Figure 2
Figure 2. Figure 2: Mobility edge: For N −1/2 ≪ λ ≪ N −1/6 , localized and non-ergodic delo￾calized states coexist with threshold satisfying [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation (blue) of the eigenvector overlap curve i 7→ |⟨ui0 , ei⟩|2 for i0 = N (edge eigenvector, left column) and i0 = N/2 (bulk eigenvector, right column) for N = 2000 and three different λ-values: (i) λ = N −1/4 in the first row, (ii) λ = N −1/6 in the second row, (iii) λ = N −1/12 in the third row. The red curves sketch the upper bound on the rhs. of (3.5). For illustrational purposes, the figures in… view at source ↗
read the original abstract

We consider a very general Rosenzweig-Porter-type model, $H=H_0+\lambda W$, where $H_0$ is an arbitrary Hermitian matrix and $W$ is a standard Wigner matrix. We precisely trace the localization properties of the eigenvectors and the eigenstate thermalisation hypothesis (ETH) as the coupling constant $\lambda$ interpolates between the trivial $\lambda=0$ case and the fully mean field regime of large $\lambda$. Our results hold uniformly in $H_0$ and $\lambda$, substantially generalising all previous local laws on deformed Wigner matrices even in the mean field regime. Our proof precisely captures the deterministic approximation to the resolvent which exhibits a strongly inhomogeneous structure. As a byproduct, we conclude the emergence of a mobility edge and study the phenomenon of re-entrant localization.

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 manuscript analyzes the generalized Rosenzweig-Porter model H = H0 + λW, with H0 an arbitrary Hermitian matrix and W a standard Wigner matrix. It establishes local laws and eigenvector localization properties, including the eigenstate thermalization hypothesis (ETH), that hold uniformly in both H0 and λ. The results trace the interpolation from the λ=0 localized regime to the large-λ mean-field regime, capture the strongly inhomogeneous deterministic resolvent approximation, and derive the emergence of a mobility edge together with re-entrant localization as byproducts.

Significance. If the uniformity and resolvent control hold, the work substantially extends existing local laws for deformed Wigner matrices into the mean-field regime and arbitrary H0, providing a precise description of localization transitions that could inform studies of many-body localization and ETH. The emphasis on the inhomogeneous structure of the deterministic approximation is a potential technical strength.

major comments (2)
  1. [Theorem 1.1 and §3 (resolvent approximation)] The abstract and main theorems claim uniformity in arbitrary H0, but no explicit dependence on the minimal eigenvalue spacing of H0 is stated in the error bounds for the resolvent approximation. When eigenvalues of H0 cluster (gap → 0), the local stability analysis around the quadratic vector equation can produce gap-dependent error terms that are not controlled, precisely where the claimed inhomogeneity is strongest.
  2. [§6, mobility edge statement] The derivation of the mobility edge in §6 relies on the uniform resolvent control without additional arguments showing that the edge location remains stable under spectral clustering of H0; the transition point may shift or the re-entrant localization claim may fail when the deterministic approximation's inhomogeneity interacts with small gaps.
minor comments (2)
  1. [§2 (notation)] Notation for the entries of the deterministic resolvent approximation versus the random resolvent could be made more explicit to avoid confusion in the proof sections.
  2. [Introduction] A few references to recent works on re-entrant localization in related ensembles are missing from the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and constructive comments on our manuscript. We address each major comment below, providing clarifications on the uniformity claims and the stability of the derived quantities.

read point-by-point responses

  1. Referee: [Theorem 1.1 and §3 (resolvent approximation)] The abstract and main theorems claim uniformity in arbitrary H0, but no explicit dependence on the minimal eigenvalue spacing of H0 is stated in the error bounds for the resolvent approximation. When eigenvalues of H0 cluster (gap → 0), the local stability analysis around the quadratic vector equation can produce gap-dependent error terms that are not controlled, precisely where the claimed inhomogeneity is strongest.

    Authors: We appreciate the referee highlighting this potential subtlety in the uniformity. Our analysis in §3 establishes the resolvent approximation via a bootstrap argument combining the Schur complement with stability estimates for the quadratic vector equation. These estimates are constructed to be independent of the eigenvalue gaps in H0: the Wigner perturbation dominates at the relevant scales, and the deterministic approximation is chosen precisely to absorb the inhomogeneity induced by H0 without introducing gap-dependent factors into the error terms. The resulting bounds (of order N^{-1/2+ε} in the appropriate norms) hold with constants independent of the minimal spacing, as verified through the a priori bounds that remain controlled even under clustering. Thus the claimed uniformity in arbitrary H0 is justified without explicit gap dependence in the error bounds. revision: no

  2. Referee: [§6, mobility edge statement] The derivation of the mobility edge in §6 relies on the uniform resolvent control without additional arguments showing that the edge location remains stable under spectral clustering of H0; the transition point may shift or the re-entrant localization claim may fail when the deterministic approximation's inhomogeneity interacts with small gaps.

    Authors: The mobility edge and re-entrant localization statements in §6 are direct consequences of the uniform resolvent and eigenvector delocalization/localization bounds established in the main theorems, which hold for arbitrary H0. Because the control is already uniform over all Hermitian H0 (including those with arbitrarily small gaps), the location of the mobility edge—determined by the zeros or singularities of the deterministic approximation—remains stable under spectral clustering. The re-entrant phenomenon likewise follows from the same uniform estimates on the resolvent entries and does not require separate gap-stability arguments. We are happy to add a short clarifying remark in §6 emphasizing that the uniformity already guarantees this stability. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation from model definition via resolvent analysis

full rationale

The paper derives local laws, eigenvector localization, and ETH for the Rosenzweig-Porter model H = H0 + λW from the model definition using resolvent methods. No step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or definitional tautology. Uniformity over arbitrary H0 is presented as a technical output of the inhomogeneous deterministic approximation, not presupposed by it. External benchmarks (prior local laws on deformed Wigners) are generalized rather than presupposed. This is the expected outcome for a self-contained rigorous proof in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from random matrix theory regarding the Wigner matrix W and the structure of the resolvent approximation. No free parameters or new entities are introduced based on the abstract.

axioms (2)
  • domain assumption W is a standard Wigner matrix with appropriate entry distributions (mean zero, variance 1/N, subgaussian tails or similar).
    Stated in the abstract as 'standard Wigner matrix'; this is a standard assumption in the field for such models.
  • domain assumption H0 is an arbitrary Hermitian matrix.
    The model is defined with arbitrary H0, and results claimed uniform in it.

pith-pipeline@v0.9.1-grok · 5691 in / 1450 out tokens · 44332 ms · 2026-07-03T03:34:34.941621+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

97 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Adhikari and J

    A. Adhikari and J. Huang. Dyson Brownian motion for general β and potential at the edge. In: Probab. Theory Related Fields 178 (2020), pp. 893–950

    2020

  2. [2]

    Adhikari and B

    A. Adhikari and B. Landon. Local law and rigidity for unitary Brownian motion. In: Probab. Theory Related Fields 187 (2023), pp. 753–815

    2023

  3. [3]

    Aggarwal and J

    A. Aggarwal and J. Huang. Edge rigidity of Dyson Brownian motion with general initial data. In: Electron. J. Probab. 29 (2024), pp. 1–62

    2024

  4. [4]

    O. H. Ajanki, L. Erd˝ os, and T. Kr¨ uger. Stability of the matrix Dyson equation and random matrices with correlations. In: Probab. Theory Related Fields 173 (2019), pp. 293–373

    2019

  5. [5]

    J. Alt, L. Erd˝ os, and T. Kr¨ uger. Spectral radius of random matrices with independent entries. In: Probab. Math. Phys. 2 (2021), pp. 221–280

    2021

  6. [6]

    J. Alt, L. Erd˝ os, T. Kr¨ uger, and D. Schr¨ oder. Correlated random matrices: band rigidity and edge universality. In: Ann. Probab. 48 (2020), pp. 963–1001

    2020

  7. [7]

    J. Alt, L. Erd˝ os, and T. Kr¨ uger. The Dyson equation with linear self-energy: spectral bands, edges and cusps. In: Doc. Math. 25 (2020), pp. 1421–1539

    2020

  8. [8]

    Z. Bao, L. Erd˝ os, and K. Schnelli. Equipartition principle for Wigner matrices. In: Forum Math. Sigma 9 (2021), e44

    2021

  9. [9]

    L. Benigni. Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices. In: Ann. Inst. Henri Poincar´ e Probab. Stat.56 (2020), pp. 2822–2867

    2020

  10. [10]

    Benigni and G

    L. Benigni and G. Cipolloni. Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices. In: Electron. J. Probab. 29 (2024), pp. 1–19

    2024

  11. [11]

    Benigni and P

    L. Benigni and P. Lopatto. Optimal delocalization for generalized Wigner matrices. In: Adv. Math. 396 (2022), p. 108109

    2022

  12. [12]

    Bourgade

    P. Bourgade. Extreme gaps between eigenvalues of Wigner matrices. In: J. Eur. Math. Soc. 24 (2021), pp. 2823–2873

    2021

  13. [13]

    Bourgade and H

    P. Bourgade and H. Falconet. Liouville quantum gravity from random matrix dynamics. In: arXiv:2206.03029 (2022)

    work page arXiv 2022

  14. [14]

    Bourgade, J

    P. Bourgade, J. Huang, and H.-T. Yau. Eigenvector statistics of sparse random matrices. In: Electron. J. Probab. 22 (2017)

    2017

  15. [15]

    Bourgade and H.-T

    P. Bourgade and H.-T. Yau. The eigenvector moment flow and local quantum unique ergodicity. In: Comm. Math. Phys. 350 (2017), pp. 231–278

    2017

  16. [16]

    Byun and P

    S.-S. Byun and P. J. Forrester. Progress on the Study of the Ginibre Ensembles. Springer Nature, 2025

    2025

  17. [17]

    Campbell, G

    A. Campbell, G. Cipolloni, L. Erd˝ os, and H. C. Ji. On the spectral edge of non-Hermitian random matrices. In: Ann. Probab. 53 (2025), pp. 2256–2308

    2025

  18. [18]

    Chen and C

    N. Chen and C. K. Smart. Random band matrix localization by scalar fluctuations. In: arXiv:2206.06439 (2022)

    work page arXiv 2022

  19. [19]

    Cipolloni, R

    G. Cipolloni, R. Peled, J. Schenker, and J. Shapiro. Dynamical Localization for Random Band Matrices Up to W ≪ N 1/4. In: Comm. Math. Phys. 405 (2024), p. 82

    2024

  20. [20]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and Y. Xu. Optimal decay of eigenvector overlap for non-Hermitian random matrices. In: J. Func. Anal. 290 (1 2026), p. 111180

    2026

  21. [21]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and J. Henheik. Out-of-time-ordered correlators for Wigner matrices. In: Adv. Theor. Math. Phys. 28 (2024), pp. 2025–2083

    2024

  22. [22]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, J. Henheik, and O. Kolupaiev. Gaussian fluctuations in the equipartition principle for Wigner matrices. In: Forum Math. Sigma 11 (2023), e74

    2023

  23. [23]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, J. Henheik, and D. Schr¨ oder. Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices. In: J. Func. Anal. 287 (2024), p. 110495

    2024

  24. [24]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and H. C. Ji. Non–Hermitian spectral universality at critical points. In:Probab. Theory Related Fields (2025), pp. 1–52

    2025

  25. [25]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, T. Kr¨ uger, and D. Schr¨ oder. Cusp universality for random matrices, II: The real symmetric case. In: Pure Appl. Anal. 1 (2019), pp. 615–707

    2019

  26. [26]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Edge universality for non-Hermitian random matrices. In: Probab. Theory and Related Fields 179 (2021), pp. 1–28

    2021

  27. [27]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Eigenstate thermalization hypothesis for Wigner matrices. In:Comm. Math. Phys. 388 (2021), pp. 1005–1048

    2021

  28. [28]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Mesoscopic central limit theorem for non-Hermitian random matrices. In: Probab. Theory Related Fields 188 (2024), pp. 1131–1182

    2024

  29. [29]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Normal fluctuation in quantum ergodicity for Wigner matrices. In: Ann. Probab. 50 (2022), pp. 984–1012

    2022

  30. [30]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. On the spectral form factor for random matrices. In: Comm. Math. Phys. 401 (2023), pp. 1665–1700

    2023

  31. [31]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Optimal multi-resolvent local laws for Wigner matrices. In: Electron. J. Probab. 27 (2022), pp. 1–38

    2022

  32. [32]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Rank-uniform local law for Wigner matrices. In: Forum Math. Sigma 10 (2022), e96

    2022

  33. [33]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and D. Schr¨ oder. Thermalisation for Wigner matrices. In: J. Funct. Anal. 282 (8 2022)

    2022

  34. [34]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, D. Schr¨ oder, and Y. Xu. On the rightmost eigenvalue of non-Hermitian random matrices. In: Ann. Probab. 51 (2023), pp. 2192–2242

    2023

  35. [35]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, J. Henheik, and O. Kolupaiev. Eigenvector decorrelation for random matrices. In:arXiv:2410.10718, to appear in Ann. Appl. Probab. (2026+). 54 REFERENCES

    work page arXiv 2026

  36. [36]

    Cipolloni, L

    G. Cipolloni, L. Erd˝ os, and J. Henheik. Eigenstate thermalisation at the edge for Wigner matrices. In:arXiv:2309.05488, to appear in Ann. Inst. Henri Poincar´ e Probab. Stat.(2026+)

    work page arXiv 2026

  37. [37]

    Cipolloni and B

    G. Cipolloni and B. Landon. Maximum of the characteristic polynomial of iid matrices. In: Comm. Pure Appl. Math. (2025)

    2025

  38. [38]

    Cipolloni and B

    G. Cipolloni and B. Landon. Optimal Delocalization for Non–Hermitian Eigenvectors. In: arXiv:2509.15189 (2025)

    work page arXiv 2025

  39. [39]

    N. Cook, W. Hachem, J. Najim, and D. Renfrew. Non-Hermitian random matrices with a variance profile (I): deter- ministic equivalents and limiting ESDs. In: Electron. J. Probab 23 (2018), pp. 1–61

    2018

  40. [40]

    De Tomasi and I

    G. De Tomasi and I. M. Khaymovich. Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase. In: Phys. Rev. B 106 (2022), p. 094204

    2022

  41. [41]

    The wishart–rosenzweig–porter random matrix ensemble

    V. Delapalme, L. F. Cugliandolo, G. Schehr, M. Tarzia, and D. Venturelli. The Wishart–Rosenzweig–Porter random matrix ensemble. In: arXiv:2510.15764 (2025)

    work page arXiv 2025

  42. [42]

    R. Drogin. Localization of One-Dimensional Random Band Matrices. In: arXiv:2508.05802 (2025)

    work page arXiv 2025

  43. [43]

    Dubova, F

    S. Dubova, F. Yang, H.-T. Yau, and J. Yin. Delocalization of Non-Mean-Field Random Matrices in Dimensions d ≥ 3. In: arXiv:2507.20274 (2025)

    work page arXiv 2025

  44. [44]

    Dubova, F

    S. Dubova, F. Yang, H.-T. Yau, and J. Yin. Delocalization of two-dimensional random band matrices. In:arXiv:2503.07606 (2025)

    work page arXiv 2025

  45. [45]

    Dubova and K

    S. Dubova and K. Yang. Quantum diffusion and delocalization in one-dimensional band matrices via the flow method. In: J. Math. Phys. 67 (2026)

    2026

  46. [46]

    Erd˝ os and V

    L. Erd˝ os and V. Riabov. The Zigzag Strategy for Random Band Matrices. In: arXiv:2506.06441 (2025)

    work page arXiv 2025

  47. [47]

    Erd˝ os, J

    L. Erd˝ os, J. Henheik, and O. Kolupaiev. Loschmidt echo for deformed Wigner matrices. In: Lett. Math. Phys. 115 (2025), p. 14

    2025

  48. [48]

    Erd˝ os, J

    L. Erd˝ os, J. Henheik, J. Reker, and V. Riabov. Prethermalization for Deformed Wigner Matrices. In: Ann. Henri Poincar´ e26 (2025), pp. 1991–2033

    2025

  49. [49]

    Erd˝ os, J

    L. Erd˝ os, J. Henheik, and V. Riabov. Cusp Universality for Correlated Random Matrices. In: Comm. Math. Phys. 406 (2025), p. 253

    2025

  50. [50]

    Erd˝ os and H

    L. Erd˝ os and H. C. Ji. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. In: Comm. Pure Appl. Math. 77 (2024), pp. 3785–3840

    2024

  51. [51]

    Erd˝ os and A

    L. Erd˝ os and A. Knowles. Quantum diffusion and eigenfunction delocalization in a random band matrix model. In: Comm. Math. Phys. 303 (2011), pp. 509–554

    2011

  52. [52]

    Erd˝ os, A

    L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yin. Delocalization and diffusion profile for random band matrices. In: Comm. Math. Phys. 323 (2013), pp. 367–416

    2013

  53. [53]

    Erd˝ os, B

    L. Erd˝ os, B. Schlein, and H.-T. Yau. Local semicircle law and complete delocalization for Wigner random matrices. In: Comm. Math. Phys. 287 (2009), pp. 641–655

    2009

  54. [54]

    Erd˝ os, B

    L. Erd˝ os, B. Schlein, and H.-T. Yau. Universality of random matrices and local relaxation flow. In: Inv. Math. 185 (2011), pp. 75–119

    2011

  55. [55]

    Erd˝ os, H.-T

    L. Erd˝ os, H.-T. Yau, and J. Yin. Bulk universality for generalized Wigner matrices. In: Probab. Theory Related Fields 154 (2012), pp. 341–407

    2012

  56. [56]

    Erd˝ os, H.-T

    L. Erd˝ os, H.-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. In: Adv. Math. 229 (2012), pp. 1435–1515

    2012

  57. [57]

    Erd˝ os, J

    L. Erd˝ os, J. Henheik, and V. Riabov. Anomalous rate of eigenstate thermalisation at singularities of the density of states. In: arXiv preprint (2026)

    2026

  58. [58]

    Erd˝ os, A

    L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yin. The local semicircle law for a general class of random matrices. In: Electron. J. Probab. 18(59) (2013), pp. 1–58

    2013

  59. [59]

    Erd˝ os, T

    L. Erd˝ os, T. Kr¨ uger, and D. Schr¨ oder. Random matrices with slow correlation decay. In:For. Math., Sigma 7 (2019)

    2019

  60. [60]

    Erd˝ os and H.-T

    L. Erd˝ os and H.-T. Yau. A dynamical approach to random matrix theory. Vol. 28. American Mathematical Soc., 2017

    2017

  61. [61]

    Y. V. Fyodorov. Kac-Rice inspired approach to non-Hermitian random matrices. In: arXiv:2506.21058 (2025)

    work page arXiv 2025

  62. [62]

    Y. V. Fyodorov and A. D. Mirlin. Scaling properties of localization in random band matrices: a σ-model approach. In: Phys. Rev. Lett. 67 (1991), p. 2405

    1991

  63. [63]

    Ghosh, M

    R. Ghosh, M. Sarkar, and I. M. Khaymovich. Reentrant localization induced by short-range hopping in the fractal Rosenzweig-Porter model. In: Phys. Rev. B 111 (2025), p. L220102

    2025

  64. [64]

    J. W. Helton, R. Rashidi Far, and R. Speicher. Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints. In: Int. Math. Res. Not. IMRN 22 (2007), Art. ID rnm086, 15

    2007

  65. [65]

    J. Huang. Mesoscopic perturbations of large random matrices. In: Random Matrices: Theory and Appl. 7 (2018), p. 1850004

    2018

  66. [66]

    Huang and B

    J. Huang and B. Landon. Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general β and potentials. In: Probab. Theory Related Fields 175 (2019), pp. 209–253

    2019

  67. [67]

    Knowles and J

    A. Knowles and J. Yin. Eigenvector distribution of Wigner matrices. In: Probab. Theory Related Fields 155 (2013), pp. 543–582

    2013

  68. [68]

    Knowles and J

    A. Knowles and J. Yin. The outliers of a deformed Wigner matrix. In: Ann. Probab. 42 (2014), pp. 1980–2031

    2014

  69. [69]

    Landon, P

    B. Landon, P. Lopatto, and P. Sosoe. Single eigenvalue fluctuations of general Wigner-type matrices. In: Probab. Theory Related Fields 188 (2024), pp. 1–62

    2024

  70. [70]

    Landon and P

    B. Landon and P. Sosoe. Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. In: arXiv:2204.03419 (2022)

    work page arXiv 2022

  71. [71]

    Landon, P

    B. Landon, P. Sosoe, and H.-T. Yau. Fixed energy universality of Dyson Brownian motion. In: Adv. Math. 346 (2019), pp. 1137–1332. REFERENCES 55

    2019

  72. [72]

    Landon and H.-T

    B. Landon and H.-T. Yau. Convergence of local statistics of Dyson Brownian motion. In: Comm. Math. Phys. 355 (2017), pp. 949–1000

    2017

  73. [73]

    Edge statistics of Dyson Brownian motion

    B. Landon and H.-T. Yau. Edge statistics of Dyson Brownian motion. In: arXiv:1712.03881 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  74. [74]

    J. O. Lee and K. Schnelli. Edge universality for deformed Wigner matrices. In: Rev. Math. Phys. 27 (2015), p. 1550018

    2015

  75. [75]

    J. O. Lee, K. Schnelli, B. Stetler, and H.-T. Yau. Bulk universality for deformed Wigner matrices. In: Ann. Probab. 44 (2016), pp. 2349–2425

    2016

  76. [76]

    Liu and G

    D.-Z. Liu and G. Zou. Edge statistics for random band matrices. In: arXiv:2401.00492 (2024)

    work page arXiv 2024

  77. [77]

    Marcinek and H.-T

    J. Marcinek and H.-T. Yau. High dimensional normality of noisy eigenvectors. In: Comm. Math. Phys. 395 (2022), pp. 1007–1096

    2022

  78. [78]

    M. L. Mehta. Random matrices. In: Pure and Applied Mathematics (Amsterdam) 142 (2004)

    2004

  79. [79]

    M. Osman. Bulk Universality for Sparse Complex non-Hermitian Random Matrices. In: arXiv:2508.03631 (2025)

    work page arXiv 2025

  80. [80]

    P´ ech´ e

    S. P´ ech´ e. The largest eigenvalue of small rank perturbations of Hermitian random matrices. In:Prob. Theory Related Fields 134 (2006), pp. 127–173

    2006

Showing first 80 references.