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REVIEW 4 major objections 5 minor 108 references

How Random Are Ergodic Eigenstates of the Ultrametric Random Matrices and the Quantum Sun Model?

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper reports that for both ultrametric random matrices and the Quantum Sun Model, the largest Schmidt eigenvalue of ergodic eigenstates, after centering and scaling, follows the extreme value distribution rather than the Tracy-Widom…

desk verdict Real MP-law deviations in UM and QSM are buried under an EVD-over-TW conclusion that the paper's own finite-size checks undercut. read the letter →

arxiv 2501.19244 v1 pith:QB5F5PE2 submitted 2025-01-31 quant-ph cond-mat.dis-nn

Tanay Pathak This is my paper · ORCID
classification quant-phcond-mat.dis-nn MSC 15B5260G7081Q5082B44
keywords extremevaluestatisticsSchmidteigenvaluesultrametricrandommatricesquantumsunmodelTracy-WidomdistributionMarchenko-Pasturlawergodiceigenstatesentanglementspectrum
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

The paper asks how random the ergodic eigenstates of two related models—ultrametric random matrices and the Quantum Sun Model—really are, and answers by looking at the extreme eigenvalues of the entanglement spectrum rather than at level spacings. For both models, the reduced density matrix built from an ergodic eigenstate should, if the eigenstate were truly random, yield Schmidt eigenvalues with the Marchenko-Pastur density and a largest eigenvalue obeying the Tracy-Widom distribution. The numerics show deviations from Marchenko-Pastur near the tail, and the centered and scaled largest Schmidt eigenvalue is fitted better by the Fisher-Tippett-Gnedenko extreme value distribution (Eq. (4)) than by Tracy-Widom, with a Weibull-type shape for the ultrametric matrices and a Fréchet-type shape for the Quantum Sun Model. The paper reads this as evidence that the Schmidt eigenvalues are much more weakly correlated than Wishart eigenvalues, so that ergodic eigenstates pass nearest-neighbor spectral statistics while still failing a stronger randomness test.

What carries the argument

The object that carries the argument is the reduced density matrix formed from an ergodic eigenstate by tracing out half the Hilbert space; its eigenvalues are the Schmidt eigenvalues. For a Haar-random or Gaussian-vector eigenstate this matrix belongs to the trace-restricted Wishart ensemble, so the paper compares the observed eigenvalue density with the Marchenko-Pastur law, the largest eigenvalue with the Tracy-Widom distribution, and the smallest eigenvalue with the exact minimum-eigenvalue formula of Eq. (19). The distinguishing mechanism is the Fisher-Tippett-Gnedenko theorem of Eq. (4), which supplies the alternative extreme value distributions that arise when the underlying variables are uncorrelated or weakly correlated; the fitted shape parameter $\tilde{\xi}$ decides whether the maximum behaves in a Weibull-like or Fréchet-like way.

What would settle it

Compute the maximum Schmidt eigenvalue distribution for trace-restricted Wishart matrices of size $2^6 \times 2^6$ using the exact finite-size mean and variance of $\lambda_{\max}$ instead of the large-$D$ asymptotic values; if the empirical distribution collapses onto the Tracy-Widom curve once exact centering is used, then the extreme-value fits reported for the ultrametric matrices and the Quantum Sun Model would need to be re-tested against that corrected Wishart baseline.

Watch

Extended reading notes

Core claim

The central claim is that the entanglement spectra of the ergodic eigenstates of ultrametric random matrices and of the Quantum Sun Model are not Wishart-like. If the eigenstates were purely random, the reduced density matrices obtained by tracing out half the $2^{12}$-dimensional Hilbert space would form trace-restricted Wishart ensembles, whose density is the Marchenko-Pastur law and whose largest eigenvalue, after centering and scaling, is Tracy-Widom distributed. The paper finds that the Schmidt eigenvalue density deviates from Marchenko-Pastur, especially near the tail, and that the distribution of the maximum Schmidt eigenvalue, centered and rescaled with the Wishart asymptotic mean and variance or with free parameters, is well described by the extreme value distribution of Eq. (4). The fitted shape parameter is positive (Weibull type) for the ultrametric matrices and negative (Fréchet type) for the Quantum Sun Model, and the minimum Schmidt eigenvalue also departs from the exact analytical distribution for trace-restricted Wishart matrices, with moment mismatch already at the first moment. The conclusion is that ergodic eigenstates in these models are not fully random: their Schmidt eigenvalues are far less correlated than those of Wishart matrices, and standard spectral statistics miss this.

Load-bearing premise

The load-bearing assumption is that the Tracy-Widom comparison is carried out with the correct centering and scaling: the paper uses the asymptotic mean and variance formulas of Eqs. (14)-(16), and its own appendix shows that even pure Wishart matrices at the sizes used here match Tracy-Widom only imperfectly, so a different finite-size centering could shift the comparison.

Editorial extensions

If this is right

  • If the extreme value distributions are the right description, then the ergodic eigenstates of ultrametric random matrices and the Quantum Sun Model fail a Tracy-Widom test even at $\alpha = 0.9$, where level spacing ratios already look random-matrix-like.
  • Nearest-neighbor spacing ratios alone are not enough to certify full randomness of ergodic eigenstates; higher-order correlation measures are needed to detect the weak correlations that extreme value statistics expose.
  • The minimum Schmidt eigenvalue distribution, which is exact for trace-restricted Wishart matrices, also deviates for both models, indicating that weak correlations extend to the lower edge of the entanglement spectrum.
  • The qualitative difference between the two models—Weibull-like for ultrametric matrices, Fréchet-like for the Quantum Sun Model—shows that the QSM is not identical to the ultrametric ensemble even in the ergodic regime.
  • Extreme value statistics of Schmidt eigenvalues can serve as a more stringent probe of eigenstate randomness in other ergodic and many-body-localized systems, as the paper argues from its two examples.

Reading between the lines

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

  • An immediate extension would be to test the same centering and scaling procedure against the exact finite-dimension mean and variance of the largest trace-restricted Wishart eigenvalue; the paper's own appendix shows that even pure Wishart matrices of size $2^6$ and $2^{12}$ do not collapse cleanly onto Tracy-Widom, so part of the reported deviation could reflect the asymptotic centering used in E
  • A testable follow-up is to generate synthetic ensembles of Schmidt eigenvalues drawn independently from the measured density and check whether their maximum eigenvalue distribution reproduces the observed extreme value fit; this would isolate how much of the effect comes from the trace constraint and how much from the Hamiltonian structure.
  • One could scan the coupling $\alpha$ across the ergodic transition and locate where the maximum eigenvalue distribution switches from extreme-value to Tracy-Widom behavior; if such a crossover exists, it would give a new finite-size diagnostic of the avalanche-to-ergodic transition.
  • The same analysis applied to disordered Heisenberg chains with variable disorder strength could connect these deviations to the many-body localization transition and to recent fading-ergodicity results, which the paper lists as a natural open direction.
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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

4 major / 5 minor

Summary. The manuscript numerically studies the Schmidt eigenvalues of reduced density matrices built from ergodic eigenstates of ultrametric random matrices (UM) and of the Quantum Sun Model (QSM), both at coupling α = 0.9. The paper reports three main findings: (i) the Schmidt eigenvalue density deviates from the Marchenko-Pastur law, especially near the tail, for both models; (ii) the centered and scaled maximum Schmidt eigenvalue does not follow the Tracy-Widom distribution but is better described by the generalized extreme value distribution (Eq. 4), which the authors interpret as evidence of weaker-than-Wishart correlations among Schmidt eigenvalues; and (iii) the minimum Schmidt eigenvalue distribution deviates from the exact trace-restricted Wishart formula of Refs. [86, 99], with stronger deviations in the QSM. The authors conclude that conventional spectral statistics such as nearest-neighbor spacing ratios are insufficient to certify full ergodicity and that extreme-value statistics provide a more stringent test.

Significance. If the main claim is correct, the paper identifies a concrete and physically interesting failure of random-matrix-type behavior in models that are otherwise considered ergodic: the maximum Schmidt eigenvalue would follow an extreme-value distribution rather than Tracy-Widom, indicating weak rather than strong eigenvalue correlations. This would be a useful diagnostic for quantum avalanche models and for the broader question of what 'randomness' of ergodic eigenstates means. The paper is also careful in some respects: it compares Marchenko-Pastur deviations against same-size Wishart matrices, uses exact minimum-eigenvalue formulas, and transparently reports that even pure Wishart matrices do not collapse perfectly onto Tracy-Widom at the simulated sizes. However, the central extreme-value claim currently rests on a three-parameter fit without a validated finite-size Wishart null, and the connection between the fitted generalized extreme-value form and weak correlations is asserted rather than tested.

major comments (4)
  1. [III.C and Appendix A (Eqs. (14)-(16), Figs. 4 and 13)] The Tracy-Widom comparison does not provide a valid finite-size null for the central claim. Equations (14)-(16) are asymptotic large-D results for the largest eigenvalue, but Fig. 4 shows that same-size (2^6 × 2^6) trace-restricted Wishart matrices already deviate visibly from Tracy-Widom, and the text in Appendix A explicitly states that even 2^12 × 2^12 pure Wishart matrices match 'still not very good'. Therefore the observed UM and QSM deviations could be entirely a finite-size centering/scaling artifact. The authors should compare against a numerical trace-restricted Wishart null at the same dimension, sample count, and analysis pipeline, or use finite-D centering and scaling. Without this, the claim that UM and QSM maxima deviate from Wishart behavior is not supported.
  2. [III.C and IV.C (Eq. (4), Figs. 5 and 11)] The positive evidence for the extreme-value distribution is a three-parameter fit (location α, scale β, shape ξ) with no reported uncertainties, no goodness-of-fit statistic, and no control experiment fitting the same generalized extreme-value family to same-size Wishart maxima. Since a three-parameter GEV family is flexible enough to approximate a broad class of skew unimodal distributions, a visually good fit for UM/QSM does not by itself distinguish weak correlations from finite-size or centering effects. The authors should report a standard goodness-of-fit test and, crucially, fit the same GEV form to the finite-size trace-restricted Wishart null; the central claim requires showing that the Wishart null does not fit GEV as well.
  3. [III.C and IV.C, Fig. 11] The QSM maximum-eigenvalue result is currently ambiguous: Section IV states that N = 5 is used for QSM, but Fig. 11's caption says the orange markers are for QSM with N = 3, and the fitting parameters quoted in Section IV.C (α = 8.15×10^-2, β = 1.13×10^-2, ξ = -2.06×10^-1) differ from the N = 3 values reported in Appendix B (α = 8.98×10^-2, β = 1.72×10^-2, ξ = -2.84×10^-1). The authors must clarify which dataset is shown in Fig. 11 and present the N = 5 result explicitly, since the QSM version of the central claim depends on this.
  4. [Section II, around Eq. (4)] The inference 'deviation from Tracy-Widom implies weaker correlations' is not justified by the Fisher-Tippett-Gnedenko theorem alone. That theorem concerns linear-normalized maxima of sequences satisfying appropriate mixing conditions, and the generalized extreme-value family can also appear as a finite-size effective description without implying true weak dependence. A concrete additional test is needed, such as checking whether the fitted shape parameter changes systematically with subsystem dimension, or directly measuring correlations among Schmidt eigenvalues, or comparing with a solvable weakly-correlated model at the same dimension.
minor comments (5)
  1. [Eqs. (3) and (12)] Equation (3) and Eq. (12) are identical presentations of the Marchenko-Pastur density; one of them should be removed or cross-referenced to avoid redundancy.
  2. [III.A and IV.A] The generalized hyperbolic distribution fits report parameters (e.g., b = 1.43068, ξ = 1.17378 in III.A; λ = 1.41909, ξ = 2.66674×10^-3 in IV.A) without stating the fitting procedure, the number of points used, or any uncertainty or goodness-of-fit measure. The text already acknowledges 'small deviations' and 'not really optimal' fits, so quantitative fit diagnostics would help.
  3. [III.C and IV.C, Figs. 4, 5, 10, and 11] The figure captions and text use inconsistent color descriptions ('blue solid line' in the text versus marker colors in the captions, and Fig. 11's N = 3 label versus the N = 5 discussion in the main text). Please make the captions self-contained and matching the text.
  4. [Table I] Table I compares moments of the minimum eigenvalue, but no statistical uncertainties are given for the numerically obtained moment values; given the very small magnitudes, error bars are important for judging whether the reported deviations (e.g., the first moment for UM) are significant.
  5. [Section IV.B, Fig. 8 caption] The caption says 'for the UM matrix case' where it should say 'for the QSM case' when describing the QSM eigenvalue distribution; please correct this copy-paste error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EVD and Tracy-Widom comparisons are empirical fits against standard external benchmarks, with no parameter fitted to data being renamed as a prediction.

full rationale

I find no circular step by the standards of this review. The paper's central claim is that the maximum Schmidt eigenvalue distribution of UM random matrices and the Quantum Sun Model, after centering and scaling, is better described by the extreme value distribution (Eq. 4) than by the Tracy-Widom distribution. This is an empirical, numerical claim: the EVD parameters (alpha, beta, xi) are fitted to the measured histogram and are reported as fit parameters, not as quantities derived from an assumption that already contains the conclusion. The Tracy-Widom comparison uses the standard asymptotic mean and variance from Eqs. (14)-(16), which are external results from random matrix theory, and the paper does not fit those quantities to the UM/QSM data. The Marchenko-Pastur and exact minimum-eigenvalue formulas also come from independent, established literature, not from the paper's own definitions. The inference from 'fits an EVD' to 'Schmidt eigenvalues are weakly correlated' is an interpretation that could be challenged, especially because the Fisher-Tippett-Gnedenko theorem is for independent variables whereas the paper observes a three-parameter fit that could approximate a broad class of skewed unimodal distributions. However, that is a statistical-reasoning concern, not circularity: no equation in the paper reduces to itself by construction, and no fitted parameter is renamed as a prediction. The paper itself reports in Appendix A that even pure Wishart matrices of size 2^12 do not collapse onto the Tracy-Widom distribution, with the match described as 'still not very good'; this undermines the finite-size null validation and is a legitimate correctness risk, but it does not make the derivation circular. There is also no load-bearing self-citation: the model constructions and the GHD fit are taken from prior work by other authors, and the paper does not invoke a uniqueness theorem from the present author's own earlier work to force its choice of distribution. For these reasons, the derivation chain is self-contained as a numerical study, and I assign a circularity score of 0.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the EVD fits, which introduce three free parameters per model, and on the null models (Marchenko-Pastur, Tracy-Widom, exact minimum eigenvalue distribution) that are standard results from random matrix theory. No new entities are postulated. The model parameters alpha=0.9, N, L are choices from prior literature rather than fitted values. The GHD parameters are fitted for a secondary observation and do not directly support the main extreme-value claim.

free parameters (10)
  • UM EVD location alpha = 6.410e-2
    Fitted so that the rescaled maximum Schmidt eigenvalue distribution matches Eq. (4) in Section III-C.
  • UM EVD scale beta = 3.822e-3
    Fitted width parameter for the extreme value fit in Section III-C.
  • UM EVD shape xi = 7.246e-2
    Fitted shape parameter; the paper interprets positive xi as Weibull-like, indicating bounded upper tail, in Section III-C.
  • QSM EVD location alpha = 8.15e-2
    Fitted for the Quantum Sun model maximum eigenvalue distribution in Section IV-C.
  • QSM EVD scale beta = 1.13e-2
    Fitted scale parameter for the QSM extreme value fit in Section IV-C.
  • QSM EVD shape xi = -2.06e-1
    Fitted shape parameter; negative xi is interpreted as Frechet-like in Section IV-C.
  • UM GHD parameter b = 1.43068
    Fitted parameter of the generalized hyperbolic distribution for eigenvector components of UM matrices in Section III-A.
  • UM GHD parameter xi = 1.17378
    Fitted parameter in the two-parameter GHD form of Eq. (11) in Section III-A.
  • QSM GHD parameter lambda = 1.41909
    Fitted parameter for QSM eigenvector distribution fit to GHD in Section IV-A.
  • QSM GHD parameter xi = 2.66674e-3
    Fitted parameter for QSM GHD fit in Section IV-A.
assumptions (6)
  • domain assumption Marchenko-Pastur law is the universal eigenvalue density for trace-restricted Wishart matrices in the large-dimension limit.
    Used in Section II as the null distribution for Schmidt eigenvalues of a random pure state.
  • standard math Tracy-Widom distribution describes the largest eigenvalue fluctuations of Wishart matrices after appropriate centering and scaling.
    Used in Section II-C as the null distribution for the maximum Schmidt eigenvalue.
  • standard math Fisher-Tippett-Gnedenko theorem applies to maxima of uncorrelated or weakly correlated random variables.
    Used in Section II to motivate the extreme value distribution as an alternative to Tracy-Widom.
  • domain assumption Alpha = 0.9 places both UM random matrices and the Quantum Sun model in the ergodic regime.
    Adopted from prior numerical studies [35] and used throughout Sections III and IV to select the parameter regime for ergodic eigenstates.
  • domain assumption The reduced density matrix of an ergodic eigenstate should be described by the trace-restricted Wishart ensemble.
    This is the null hypothesis being tested in Section II-B and III-B.
  • domain assumption The generalized hyperbolic distribution is a suitable fitting form for eigenvector component distributions of extended states.
    Used in Section III-A following the conjecture of Bogomolny and Sieber (2018).

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Cite this review

Pith. "Pith review of How Random Are Ergodic Eigenstates of the Ultrametric Random Matrices and the Quantum Sun Model?." pith.science (2026). https://pith.science/paper/QB5F5PE2

@misc{pith2026250119244,
  author       = {Pith},
  title        = {Pith review of: How Random Are Ergodic Eigenstates of the Ultrametric Random Matrices and the Quantum Sun Model?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QB5F5PE2}},
  note         = {Machine review of arXiv:2501.19244}
}
read the original abstract

We numerically study the extreme-value statistics of the Schmidt eigenvalues of reduced density matrices obtained from the ergodic eigenstates. We start by exploring the extreme value statistics of the ultrametric random matrices and then the related Quantum Sun Model, which is also a toy model of avalanche theory. It is expected that these ergodic eigenstates are purely random and thus possess random matrix theory-like features, and the corresponding eigenvalue density should follow the universal Marchenko-Pastur law. Nonetheless, we find deviations, specifically near the tail in both cases. Similarly, the distribution of maximum eigenvalue, after appropriate centering and scaling, should follow the Tracy-Widom distribution. However, our results show that, for both the ultrametric random matrix and the Quantum Sun model, it can be better described using the extreme value distribution. As the extreme value distribution is associated with uncorrelated or weakly correlated random variables, the results hence indicate that the Schmidt eigenvalues exhibit much weaker correlations compared to the strong correlations typically observed in Wishart matrices. Similar deviations are observed for the case of minimum Schmidt eigenvalues as well . Despite the spectral statistics, such as nearest neighbor spacing ratios, aligning with the random matrix theory predictions, our findings reveal that randomness is still not fully achieved. This suggests that deviations in extreme-value statistics offer a stringent test to probe the randomness of ergodic eigenstates and can provide deeper insights into the underlying structure and correlations in ergodic systems.

Figures

Figures reproduced from arXiv: 2501.19244 by the authors.

Figure 1
Figure 1. FIG. 1. The top figure shows the distribution of rescaled [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Distribution of rescaled eigenvalues, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Distribution of maximum Schmidt eigenvalue, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Distribution of minimum Schmidt eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Distribution of rescaled eigenvalues, [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (Top) Distribution of rescaled eigenvector compo [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of moments, [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Distribution of maximum Schmidt eigenvalue, [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Distribution of minimum Schmidt eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Distribution of rescaled eigenvalues, [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Distribution of [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (Top) Distribution of rescaled eigenvector compo [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Distribution of [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. (Top) Distribution of rescaled eigenvalues, [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. (Top) Distribution of maximum Schmidt eigenvalue, [PITH_FULL_IMAGE:figures/full_fig_p012_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Distribution of minimum Schmidt eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p013_18.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.