Pith. sign in

REVIEW 4 minor 113 references

In the fractal phase of generalized Rosenzweig–Porter models, level counting statistics around the Thouless energy collapse to one universal scaling function shared by all the variants studied.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 02:57 UTC pith:MOD7PG3Y

load-bearing objection Solid extension of the RP full-counting-statistics calculation to free and Wigner classes, with clean numerics and a useful QREM contrast.

arxiv 2607.09444 v1 pith:MOD7PG3Y submitted 2026-07-10 cond-mat.dis-nn cond-mat.stat-mech

Level statistics in the fractal phase of generalized Rosenzweig--Porter models

Victor Delapalme , Leticia F. Cugliandolo , Alexander K. Hartmann , Marco Tarzia , Davide Venturelli This is my paper
classification cond-mat.dis-nn cond-mat.stat-mech
keywords Rosenzweig-Porter modelfractal eigenstateslevel compressibilityThouless energyfull counting statisticsfree probabilityreplica methodquantum random energy model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The Rosenzweig–Porter ensemble is a simple random-matrix toy for the crossover from quantum chaos to localization. Between the fully extended and fully localized regimes sits a fractal phase whose eigenstates occupy only a vanishing fraction of Hilbert space. This paper generalizes the off-diagonal piece of the model to two broad classes (rotationally invariant matrices and Wigner matrices) and computes the full counting statistics of eigenvalues inside an energy window of width set by the Thouless energy. Using free probability and the replica method it shows that the cumulant generating function collapses onto a single, closed-form scaling function that is insensitive to the microscopic distributions of both matrices. Exact diagonalization of large samples and large-deviation sampling down to probabilities of order 10^{-40} confirm the analytic form. When the same observable is measured in the quantum random energy model, a scaling collapse still appears but the shape is different; the authors attribute the discrepancy to multifractal rather than simply fractal eigenstates. The result therefore both sharpens the random-matrix description of the intermediate phase and marks its limitations for more realistic many-body systems.

Core claim

In the fractal phase 1<γ<2 the cumulant generating function of the number of eigenvalues lying in an interval of width of order the Thouless energy ET takes the universal scaling form F(s)=pa(0)ET G(s,E/ET)+O(σ^{2}), where the function G is independent of the distributions of the diagonal matrix A and of the off-diagonal matrix M (provided M is either orthogonally invariant or a finite-variance Wigner matrix).

What carries the argument

The replica path-integral representation of the cumulant generating function, reduced by free-probability (HCIZ) techniques or by direct averaging over independent entries, to a saddle-point action that at leading order in the Thouless scale depends only on the second free (or ordinary) cumulant of M.

Load-bearing premise

The calculation keeps only the second cumulant of the off-diagonal matrix and discards higher-order terms; this is controlled only when the energy window is of order the Thouless energy and the matrix entries have finite variance.

What would settle it

Compute the level compressibility of a generalized RP matrix whose off-diagonal entries have infinite variance (or of a model with strongly correlated diagonal entries) at the Thouless scale and check whether it still collapses onto the same analytic curve χT(y).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the full counting statistics of eigenvalues in generalized Rosenzweig–Porter ensembles H = A + √σ M, where A is diagonal with i.i.d. entries and M is either orthogonally invariant (mutually free with A) or a finite-variance Wigner matrix. In the fractal phase 1 < γ < 2, a replica path-integral calculation combined with free-probability tools shows that the cumulant generating function of the number of eigenvalues in an interval of width E ∼ E_T takes the universal scaling form F[−E,E](s) = p_a(0) E_T G(s, E/E_T) + O(σ²), with G given explicitly by Eq. (78). The associated level compressibility χ_T(y) (Eq. 85) is likewise universal. The analytic predictions are checked by exact diagonalization (N up to 3·10^4) across several ensembles and by large-deviation sampling of the rate function down to probabilities ∼10^{-40}. A numerical contrast with the quantum random energy model shows a different, broader crossover, attributed to multifractality of the QREM eigenstates.

Significance. The work supplies a controlled analytic derivation of a previously conjectured universal scaling function for the level compressibility and full counting statistics of RP-type models. The derivation unifies the GOE, Wishart and finite-variance Wigner cases under a single saddle-point action that retains only the second free (or ordinary) cumulant of M, and recovers earlier results as special cases. High-quality numerical support—including large-deviation sampling to 10^{-40}—and a clear negative result for the QREM strengthen the claim that the universality class is tied to simple fractal (rather than multifractal) eigenstates. The result is a useful benchmark for random-matrix models of the integrability-to-chaos crossover.

minor comments (4)
  1. In §5.1 the finite-size drift of Φ(0) is left open (logarithmic growth versus possible saturation). A short remark clarifying that the analytic prediction Φ(k)∼2/k as k→0 is expected only for N→∞, and that the observed slow drift is consistent with the independent-level benchmark of Appendix B, would help the reader.
  2. Eq. (48) defines E_T with the second cumulant of M; it would be useful to state explicitly in the caption of Fig. 3 how this prefactor is fixed for each ensemble (especially the sparse-Gaussian and tent cases) so that the collapse is fully reproducible.
  3. The discussion of Lévy matrices in §5.4 notes that they empirically obey the same χ_T even though the finite-variance assumption is violated. A one-sentence pointer to the density–density correlator of Ref. [59] would make the connection sharper.
  4. Typographical: the arXiv date stamp “13 July 2026” and a few missing spaces around inline math (e.g. “1<γ<2”) should be cleaned before final production.

Circularity Check

0 steps flagged

No significant circularity: the universal CGF scaling is obtained from a controlled saddle-point evaluation of the replica action, recovering prior special cases rather than assuming them.

full rationale

The central claim is the universal form of the cumulant generating function F[−E,E](s)=pa(0)ET G(s,E/ET)+O(σ²) with G given by Eq. (78) for both orthogonally invariant and finite-variance Wigner matrices in the fractal phase. Sections 3.1–3.2 construct the replica path-integral actions for the two classes and show they coincide at O(σ) once only the second free (or ordinary) cumulant of M is retained; the subsequent saddle-point evaluation in §4.1 then produces a scaling function independent of pa and of all higher cumulants of M. The O(σ²) truncation is self-consistently justified for E∼ET≪1. Prior results of overlapping authors (the GOE case of Ref. [1] and the Wishart case of Ref. [66]) are recovered as special cases of the same calculation rather than used as load-bearing premises. The Thouless energy is the standard definition already present in the RP literature; it is not fitted to the data being predicted. Numerical validation (exact diagonalization at N=3·10⁴ and large-deviation sampling to 10^{-40}) is independent of the analytic derivation. The QREM contrast is presented as a negative result outside the claimed class. The only mild self-reference is the recovery of earlier special cases, which does not force the general result. Hence the derivation is self-contained against its own inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard free-probability and replica technology plus the modeling assumption that M has finite second cumulant and is either free or entry-wise independent of A. No new particles or forces are postulated; the only free parameters are the conventional RP control parameters γ and ν that define the phase diagram.

free parameters (3)
  • γ (RP exponent)
    Controls the variance of off-diagonal elements and selects the fractal phase 1<γ<2; chosen by hand to sit inside that phase (typically γ=1.5).
  • ν (overall scale of M)
    Sets the absolute magnitude of the Thouless energy; fixed to O(1) values (usually ν=1) without fitting.
  • effective fractal dimension D (QREM only)
    Extracted by requiring best collapse of χ(ω) curves for different system sizes; used only for the QREM comparison, not for the RP universality claim.
axioms (4)
  • standard math Low-rank HCIZ integral approximation for orthogonally invariant matrices (Eq. 23)
    Standard free-probability result used to average over M when rank(T)≪N.
  • domain assumption Replica-symmetric saddle point for the auxiliary matrix Q
    Assumed by continuity with the GOE calculation of Ref. [1]; not proved to be the unique extremum.
  • domain assumption Analytic continuation n±→±is/π of the replica partition function
    Standard but non-rigorous step of the replica method; justified a posteriori by numerical agreement.
  • domain assumption Finite second cumulant of the entries of M (κ₂^{(ζ)} or κ₂^{(free)})
    Required for the expansion of the action to O(σ); excludes Lévy matrices (which nevertheless empirically obey the same scaling).

pith-pipeline@v1.1.0-grok45 · 42030 in / 2542 out tokens · 41091 ms · 2026-07-13T02:57:06.457456+00:00 · methodology

0 comments
read the original abstract

The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy $E_T$, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around $E_T$, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as $10^{-40}$. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.

Figures

Figures reproduced from arXiv: 2607.09444 by Alexander K. Hartmann, Davide Venturelli, Leticia F. Cugliandolo, Marco Tarzia, Victor Delapalme.

Figure 1
Figure 1. Figure 1: The phase diagram of the GRP model (see Section 2). the local chaotic or non-chaotic behavior of the system. This issue is conventionally addressed through the unfolding procedure, which normalizes the spectrum to enforce a uniform mean level spacing, or more conveniently by considering the ratios of consecutive spacings [63], defined as rn = min(δn, δn+1)/ max(δn, δn+1), which are by construction independ… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The distribution of the full counting statistics IN [−E, E] for W = √ 3, ν = 1, γ = 1.5, E = 3ET , and matrix sizes N = 20, 50, 100, 200, 500, 1000 and 2000. (b) The corresponding shifted rate function Φ(k) − Φmin as a function of the counting statistics k = IN /N pa(0)ET , rescaled according to Eq. (50). The solid line shows the theoretical prediction, while the dashed line the rate function obtained … view at source ↗
Figure 3
Figure 3. Figure 3: Level compressibility χ(E) for (a) the Symmetrized Wishart–RP model, and (b) for Wigner–RP matrices. Symbols denote data from exact numerical diagonalization of random sample matrices of size N = 30000. For energies of the scales of the Thouless energy ET (48), all curves collapse on the universal prediction χ(E) = χT (E/ET ) given in Eq. (85) and drawn with a dash-dotted line. In (b), the first three sets… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic out-of-equilibrium phase diagram of the QREM (see also Refs. [76–81]). The transition line separating the fully ergodic phase from the intermediate partially delocalized, non-ergodic phase (blue), and the transition line separating the intermediate phase from the localized phase (red), are obtained from the numerical solution of Eqs. (104), (105), and (106) (see Ref. [79] for details). The dashed… view at source ↗
Figure 5
Figure 5. Figure 5: Level compressibility χ(ω) for the QREM, Eq. (96), at Γ = 0.4 for (a) ε = 0.18, (b) ε = 0.24, and (c) ε = 0.28, spanning the intermediate non-ergodic delocalized phase of the model (see [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Fractal dimensions D1 (blue) and D2 (red) of the QREM eigenstates for Γ = 0.4 and |ε| = 0.28, computed numerically from Eq. (110). The horizontal black dotted line indicates the value of the fractal exponent D that yields the best collapse of the level compressibility data for different system sizes (see [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗

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

113 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Venturelli D, Cugliandolo L F, Schehr G and Tarzia M 2023SciPost Phys.14110 URL https://scipost.org/10.21468/SciPostPhys.14.5.110

    work page doi:10.21468/scipostphys.14.5.110

  2. [2]

    Gornyi I V, Mirlin A D and Polyakov D G 2005Phys. Rev. Lett.95(20) 206603 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.95.206603 Level statistics in the fractal phase of generalized Rosenzweig–Porter models41

    doi:10.1103/physrevlett.95.206603

  3. [3]

    Phys.3211126–1205 URLhttps: //doi.org/10.1016/j.aop.2005.11.014

    Basko D M, Aleiner I L and Altshuler B L 2006Ann. Phys.3211126–1205 URLhttps: //doi.org/10.1016/j.aop.2005.11.014

    doi:10.1016/j.aop.2005.11.014 2005

  4. [4]

    Altman E and Vosk R 2015Annu. Rev. Condens. Matter Phys.6383–409 ISSN 1947-5462 URL http://dx.doi.org/10.1146/annurev-conmatphys-031214-014701

    doi:10.1146/annurev-conmatphys-031214-014701 1947

  5. [5]

    Nandkishore R and Huse D A 2015Annu. Rev. Condens. Matter Phys.615–38 ISSN 1947-5462 URLhttp://dx.doi.org/10.1146/annurev-conmatphys-031214-014726

    doi:10.1146/annurev-conmatphys-031214-014726 1947

  6. [6]

    Abanin D A and Papić Z 2017Ann. Phys. Berlin529ISSN 1521-3889 URLhttp://dx.doi. org/10.1002/andp.201700169

    doi:10.1002/andp.201700169

  7. [7]

    Alet F and Laflorencie N 2018C. R. Phys.19498–525 ISSN 1631-0705 URLhttp://dx.doi. org/10.1016/j.crhy.2018.03.003

    doi:10.1016/j.crhy.2018.03.003 2018

  8. [8]

    Abanin D A, Altman E, Bloch I and Serbyn M 2019Rev. Mod. Phys.91(2) 021001 URL https://link.aps.org/doi/10.1103/RevModPhys.91.021001

    doi:10.1103/revmodphys.91.021001

  9. [9]

    Sierant P, Lewenstein M, Scardicchio A, Vidmar L and Zakrzewski J 2025Rep. Prog. Phys.88 026502 URLhttp://dx.doi.org/10.1088/1361-6633/ad9756

    doi:10.1088/1361-6633/ad9756

  10. [10]

    Cugliandolo L F and Müller M 2023 Quantum glasses - a reviewSpin Glass Theory and Far Beyond - Replica Symmetry Breaking after 40 yearsed Charbonneau P, Marinari E, Mézard M, Parisi G, Ricci-Tersenghi F, Sicuro G and Zamponi F (World Scientific) p 353–374 URL https://www.worldscientific.com/doi/abs/10.1142/9789811273926_0018

    work page doi:10.1142/9789811273926_0018 2023

  11. [11]

    Srednicki M 1994Phys. Rev. E50(2) 888–901 URLhttps://link.aps.org/doi/10.1103/ PhysRevE.50.888

  12. [12]

    Rigol M, Dunjko V and Olshanii M 2008Nature452854–858 URLhttps://doi.org/10.1038/ nature06838

  13. [13]

    Livan G, Novaes M and Vivo P 2018Introduction to Random Matrices(Springer International Publishing) URLhttps://doi.org/10.1007/978-3-319-70885-0

    doi:10.1007/978-3-319-70885-0

  14. [14]

    Potters M and Bouchaud J P 2020A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists(CambridgeUniversityPress)URLhttps://doi.org/10.1017/ 9781108768900

  15. [15]

    Mingo J A and Speicher R 2017Free Probability and Random Matrices(Heidelberg: Springer) URLhttps://doi.org/10.1007/978-1-4939-6942-5

    doi:10.1007/978-1-4939-6942-5

  16. [16]

    CrapsB,ClerckMD,EvninOandPavlovM2026Randommatrixtheoryofintegrability-to-chaos transition (Preprint2604.03669) URLhttps://arxiv.org/abs/2604.03669

    Pith/arXiv arXiv

  17. [17]

    Mirlin A D, Fyodorov Y V, Dittes F M, Quezada J and Seligman T H 1996Phys. Rev. E54(4) 3221–3230 URLhttps://link.aps.org/doi/10.1103/PhysRevE.54.3221

    work page doi:10.1103/physreve.54.3221

  18. [18]

    Fyodorov Y V, Ossipov A and Rodriguez A 2009J. Stat. Mech.2009L12001 URLhttps: //doi.org/10.1088/1742-5468/2009/12/L12001

    work page doi:10.1088/1742-5468/2009/12/l12001 2009

  19. [19]

    Roy S, Khaymovich I M, Das A and Moessner R 2018SciPost Phys.4025 URLhttps: //scipost.org/10.21468/SciPostPhys.4.5.025

    work page doi:10.21468/scipostphys.4.5.025

  20. [20]

    Wang J, Liu X J, Xianlong G and Hu H 2016Phys. Rev. B93(10) 104504 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.93.104504

    work page doi:10.1103/physrevb.93.104504

  21. [21]

    Nosov P A, Khaymovich I M and Kravtsov V E 2019Phys. Rev. B99(10) 104203 URL https://link.aps.org/doi/10.1103/PhysRevB.99.104203

    work page doi:10.1103/physrevb.99.104203

  22. [22]

    Duthie A, Roy S and Logan D E 2022Phys. Rev. B106(2) L020201 URLhttps://link.aps. org/doi/10.1103/PhysRevB.106.L020201

    work page doi:10.1103/physrevb.106.l020201

  23. [23]

    21468/SciPostPhys.11.6.101

    Kutlin A G and Khaymovich I M 2021SciPost Phys.11101 URLhttps://scipost.org/10. 21468/SciPostPhys.11.6.101

  24. [24]

    Motamarri V R, Gorsky A S and Khaymovich I M 2022SciPost Phys.13117 URLhttps: //scipost.org/10.21468/SciPostPhys.13.5.117

    work page doi:10.21468/scipostphys.13.5.117

  25. [25]

    Tang W and Khaymovich I M 2022Quantum6733 URLhttps://doi.org/10.22331/ q-2022-06-09-733

    2022

  26. [26]

    Tarzia M 2022Phys. Rev. B105(17) 174201 URLhttps://link.aps.org/doi/10.1103/ PhysRevB.105.174201 Level statistics in the fractal phase of generalized Rosenzweig–Porter models42

  27. [27]

    Skvortsov M A, Amini M and Kravtsov V E 2022Phys. Rev. B106(5) 054208 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.106.054208

    doi:10.1103/physrevb.106.054208

  28. [28]

    Cai X, Lang L J, Chen S and Wang Y 2013Phys. Rev. Lett.110(17) 176403 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.110.176403

    work page doi:10.1103/physrevlett.110.176403

  29. [29]

    DeGottardi W, Sen D and Vishveshwara S 2013Phys. Rev. Lett.110(14) 146404 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.110.146404

    work page doi:10.1103/physrevlett.110.146404

  30. [30]

    Liu F, Ghosh S and Chong Y D 2015Phys. Rev. B91(1) 014108 URLhttps://link.aps.org/ doi/10.1103/PhysRevB.91.014108

    doi:10.1103/physrevb.91.014108

  31. [31]

    Das A K and Ghosh A 2022Phys. Rev. E105(5) 054121 URLhttps://link.aps.org/doi/10. 1103/PhysRevE.105.054121

  32. [32]

    Ahmed A, Ramachandran A, Khaymovich I M and Sharma A 2022Phys. Rev. B106(20) 205119 URLhttps://link.aps.org/doi/10.1103/PhysRevB.106.205119

    work page doi:10.1103/physrevb.106.205119

  33. [33]

    Lee S, Andreanov A and Flach S 2023Phys. Rev. B107(1) 014204 URLhttps://link.aps. org/doi/10.1103/PhysRevB.107.014204

    work page doi:10.1103/physrevb.107.014204

  34. [34]

    Das A K and Ghosh A 2023J. Phys. A: Math. Theor.56495003 ISSN 1751-8121 URL http://dx.doi.org/10.1088/1751-8121/ad0b5a

    doi:10.1088/1751-8121/ad0b5a

  35. [35]

    Le Doussal P 2025J. Phys. A: Math. Theor.ISSN 1751-8121 URLhttp://dx.doi.org/10. 1088/1751-8121/adf835

  36. [36]

    Baron J W 2025EPL15121002 ISSN 1286-4854 URLhttp://dx.doi.org/10.1209/ 0295-5075/adf2dc

  37. [37]

    Šuntajs J and Vidmar L 2022Phys. Rev. Lett.129(6) 060602 URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.129.060602

    doi:10.1103/physrevlett.129.060602

  38. [38]

    Phys.17122002 URL https://doi.org/10.1088/1367-2630/17/12/122002

    Kravtsov V E, Khaymovich I M, Cuevas E and Amini M 2015New J. Phys.17122002 URL https://doi.org/10.1088/1367-2630/17/12/122002

    work page doi:10.1088/1367-2630/17/12/122002

  39. [39]

    von Soosten P and Warzel S 2019Lett. Math. Phys.109905–922 ISSN 1573-0530 URL https://doi.org/10.1007/s11005-018-1131-7

    work page doi:10.1007/s11005-018-1131-7

  40. [40]

    Lett.11547003 URLhttps://doi.org/10

    Facoetti D, Vivo P and Biroli G 2016Europhys. Lett.11547003 URLhttps://doi.org/10. 1209/0295-5075/115/47003

  41. [41]

    Lett.11637002 URLhttps://doi.org/10.1209/ 0295-5075/116/37002

    Truong K and Ossipov A 2016Europhys. Lett.11637002 URLhttps://doi.org/10.1209/ 0295-5075/116/37002

  42. [42]

    Bogomolny E and Sieber M 2018Phys. Rev. E98(3) 032139 URLhttps://link.aps.org/doi/ 10.1103/PhysRevE.98.032139

    doi:10.1103/physreve.98.032139

  43. [43]

    Tomasi G D, Amini M, Bera S, Khaymovich I M and Kravtsov V E 2019SciPost Phys.6(1) 14 URLhttps://scipost.org/10.21468/SciPostPhys.6.1.014

    doi:10.21468/scipostphys.6.1.014

  44. [44]

    Lett.11730003 URLhttps://doi.org/10.1209/0295-5075/117/ 30003

    Amini M 2017Europhys. Lett.11730003 URLhttps://doi.org/10.1209/0295-5075/117/ 30003

    work page doi:10.1209/0295-5075/117/

  45. [45]

    Pino M, Tabanera J and Serna P 2019J. Phys. A: Math. Theor.52475101 URLhttps: //doi.org/10.1088/1751-8121/ab4b76

    work page doi:10.1088/1751-8121/ab4b76

  46. [46]

    Berkovits R 2020Phys. Rev. B102(16) 165140 URLhttps://link.aps.org/doi/10.1103/ PhysRevB.102.165140

  47. [47]

    Kravtsov V E, Khaymovich I M, Altshuler B L and Ioffe L B 2020preprint(Preprint2002.02979) URLhttps://doi.org/10.48550/arXiv.2002.02979

    Pith/arXiv arXiv doi:10.48550/arxiv.2002.02979 2002

  48. [48]

    Khaymovich I M, Kravtsov V E, Altshuler B L and Ioffe L B 2020Phys. Rev. Res.2(4) 043346 URLhttps://link.aps.org/doi/10.1103/PhysRevResearch.2.043346

    work page doi:10.1103/physrevresearch.2.043346

  49. [49]

    Monthus C 2017J. Phys. A: Math. Theor.50295101 URLhttps://iopscience.iop.org/ article/10.1088/1751-8121/aa77e1

    doi:10.1088/1751-8121/aa77e1

  50. [50]

    Biroli G and Tarzia M 2021Phys. Rev. B103(10) 104205 URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.103.104205

    work page doi:10.1103/physrevb.103.104205

  51. [51]

    Buijsman W and Lev Y B 2022SciPost Phys.12082 URLhttps://scipost.org/10.21468/ SciPostPhys.12.3.082

  52. [52]

    Level statistics in the fractal phase of generalized Rosenzweig–Porter models43 21468/SciPostPhys.11.2.045

    Khaymovich I M and Kravtsov V E 2021SciPost Phys.11045 URLhttps://scipost.org/10. Level statistics in the fractal phase of generalized Rosenzweig–Porter models43 21468/SciPostPhys.11.2.045

  53. [53]

    Sarkar M, Ghosh R and Khaymovich I M 2023Phys. Rev. B108(6) L060203 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.108.L060203

    work page doi:10.1103/physrevb.108.l060203

  54. [54]

    Buijsman W 2024Phys. Rev. B109(2) 024205 URLhttps://link.aps.org/doi/10.1103/ PhysRevB.109.024205

  55. [55]

    Phys.26083018 ISSN 1367-2630 URLhttp://dx.doi.org/10.1088/1367-2630/ad5d86

    Čadež T, Kumar Nandy D, Rosa D, Andreanov A and Dietz B 2024New J. Phys.26083018 ISSN 1367-2630 URLhttp://dx.doi.org/10.1088/1367-2630/ad5d86

    work page doi:10.1088/1367-2630/ad5d86

  56. [56]

    Jahnke V, Nandy P, Pal K, Camargo H A and Kim K Y 2025 Free probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles (Preprint2506.04520) URLhttps://arxiv.org/abs/2506.04520

    arXiv 2025

  57. [57]

    Ghosh R, Sarkar M and Khaymovich I M 2025Phys. Rev. B111(22) L220102 URLhttps: //link.aps.org/doi/10.1103/k957-fcmh

    work page doi:10.1103/k957-fcmh

  58. [58]

    Lunkin A and Tikhonov K 2025SciPost Phys.19015 URLhttps://scipost.org/10.21468/ SciPostPhys.19.1.015

  59. [59]

    org/10.21468/SciPostPhys.20.1.003

    Safonova E, Lunkin A and Feigel’man M 2026SciPost Phys.20003 URLhttps://scipost. org/10.21468/SciPostPhys.20.1.003

    work page doi:10.21468/scipostphys.20.1.003

  60. [60]

    Świętek R, Kliczkowski M, Hopjan M and Vidmar L 2026 Fading ergodicity and quantum dynamics in random matrix ensembles (Preprint2603.23616) URLhttps://arxiv.org/abs/ 2603.23616

    arXiv 2026

  61. [61]

    21468/SciPostPhys.16.1.008

    Kutlin A and Khaymovich I M 2024SciPost Phys.16008 URLhttps://scipost.org/10. 21468/SciPostPhys.16.1.008

  62. [62]

    Rev.120(5) 1698–1714 URLhttps://link.aps.org/ doi/10.1103/PhysRev.120.1698

    Rosenzweig N and Porter C E 1960Phys. Rev.120(5) 1698–1714 URLhttps://link.aps.org/ doi/10.1103/PhysRev.120.1698

    doi:10.1103/physrev.120.1698

  63. [63]

    Atas Y Y, Bogomolny E, Giraud O and Roux G 2013Phys. Rev. Lett.110(8) 084101 URL https://link.aps.org/doi/10.1103/PhysRevLett.110.084101

    doi:10.1103/physrevlett.110.084101

  64. [64]

    Rep.326259–382 ISSN 0370-1573 URLhttps://www.sciencedirect

    Mirlin A D 2000Phys. Rep.326259–382 ISSN 0370-1573 URLhttps://www.sciencedirect. com/science/article/pii/S0370157399000915

  65. [65]

    Phys.456169300 ISSN 0003-4916 URLhttps://www

    Altshuler B and Kravtsov V 2023Ann. Phys.456169300 ISSN 0003-4916 URLhttps://www. sciencedirect.com/science/article/pii/S0003491623000854

  66. [66]

    Delapalme V, Cugliandolo L F, Schehr G, Tarzia M and Venturelli D 2026J. Stat. Mech.2026 013301 ISSN 1742-5468 URLhttp://dx.doi.org/10.1088/1742-5468/ae29f6

    work page doi:10.1088/1742-5468/ae29f6 2026

  67. [67]

    Cipolloni G, Erdős L and Henheik J 2026 On a Rosenzweig-Porter-type model (Preprint 2607.02446) URLhttps://arxiv.org/abs/2607.02446

    Pith/arXiv arXiv 2026

  68. [68]

    Voiculescu D V, Dykema K J and Nica A 1992Free random variablesCRM Monograph Series (American Mathematical Soc.) ISBN 9780821869703 URLhttps://doi.org/10.1090/crmm/ 001

    work page doi:10.1090/crmm/

  69. [69]

    Vallini E, Foini L and Pappalardi S 2025 Refinements of the eigenstate thermalization hypothesis under local rotational invariance via free probability (Preprint2511.23217) URLhttps: //arxiv.org/abs/2511.23217

    Pith/arXiv arXiv 2025

  70. [70]

    Camargo H A, Fu Y, Jahnke V, Pal K and Kim K Y 2025 Quantum signatures of chaos from free probability (Preprint2503.20338) URLhttps://arxiv.org/abs/2503.20338

    arXiv 2025

  71. [71]

    WishartJ1928Biometrika20A32ISSN0006-3444URLhttp://dx.doi.org/10.2307/2331939

    work page doi:10.2307/2331939

  72. [72]

    Phys.456169300 URLhttps://www.sciencedirect

    Altshuler B and Kravtsov V 2023Ann. Phys.456169300 URLhttps://www.sciencedirect. com/science/article/pii/S0003491623000854

  73. [73]

    Cohen D, Yukalov V I and Ziegler K 2016Phys. Rev. A93042101 URLhttps://doi.org/10. 1103/PhysRevA.93.042101

  74. [74]

    Adler D, Wei D, Will M, Srakaew K, Agrawal S, Weckesser P, Moessner R, Pollmann F, Bloch I and Zeiher J 2024Nature63680 URLhttps://doi.org/10.1038/s41586-024-08188-0

    doi:10.1038/s41586-024-08188-0

  75. [75]

    Phys.8172 URL https://doi.org/10.1038/s42005-025-02039-8

    Jeyaretnam J, Bhore T, Osborne J J, Halimeh J C and Papić Z 2025Commun. Phys.8172 URL https://doi.org/10.1038/s42005-025-02039-8

    doi:10.1038/s42005-025-02039-8

  76. [76]

    Faoro L, Feigel’man M V and Ioffe L 2019Ann. Phys.409167916 ISSN 0003-4916 URL Level statistics in the fractal phase of generalized Rosenzweig–Porter models44 https://www.sciencedirect.com/science/article/pii/S000349161930171X

  77. [77]

    Baldwin C L and Laumann C R 2018Phys. Rev. B97(22) 224201 URLhttps://link.aps. org/doi/10.1103/PhysRevB.97.224201

    work page doi:10.1103/physrevb.97.224201

  78. [78]

    Parolini T and Mossi G 2020preprint(Preprint2007.00315) URLhttps://doi.org/10.48550/ arXiv.2007.00315

    Pith/arXiv arXiv 2007

  79. [79]

    Biroli G, Facoetti D, Schiró M, Tarzia M and Vivo P 2021Phys. Rev. B103(1) 014204 URL https://link.aps.org/doi/10.1103/PhysRevB.103.014204

    work page doi:10.1103/physrevb.103.014204

  80. [80]

    KechedzhiK,SmelyanskiyV,McCleanJR,DenchevVS,MohseniM,IsakovS,BoixoS,Altshuler B and Neven H 2018Leibniz Int. Proc. Inf.1119:1–9:16 URLhttps://drops.dagstuhl.de/ storage/00lipics/lipics-vol111-tqc2018/LIPIcs.TQC.2018.9/LIPIcs.TQC.2018.9.pdf

    2018

Showing first 80 references.