Pith. sign in

REVIEW 2 major objections 5 minor 71 references

Energy dynamics in a class of local random matrix Hamiltonians

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read For local random-matrix Hamiltonians with square-one bonds, the large-dimension energy dynamics reduces to a single free particle hopping on a lattice.

desk verdict The two-term large-q mapping is a genuinely exact result and the best part of the paper; the z>=3 and extended-chain results are plausible but lean on an imported vanishing claim that is checked, not proven. read the letter →

arxiv 2502.05045 v3 pith:MWFRFMFJ submitted 2025-02-07 cond-mat.stat-mech cond-mat.str-elhep-thquant-ph

Klée Pollock , Jonathan D. Kroth , Nathan Pagliaroli , Thomas Iadecola , Jonathon Riddell This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.str-elhep-thquant-ph
keywords randommatrixtheoryenergytransportlocalHamiltoniansHaar-Isingmodelsingle-particlemappingBethelatticefreeprobabilityout-of-time-orderedcorrelator
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 studies Hamiltonians built from nearest-neighbor random matrix bonds $h_j$ that square to one and are Haar-randomly rotated. Its central claim is that in the limit of large local Hilbert space dimension $q$, every single-trace correlation function of these bonds reduces to the dynamics of one free particle: the alternating correlators $\langle(h_0h_1)^m\rangle$ vanish as $O(q^{-2})$, so only words that cancel completely survive. This yields exact formulas for the density of states, the energy autocorrelator $C(t)=\frac{1}{2}+J_1(4t)/(4t)$, an out-of-time-ordered correlator, and the finite-temperature response of a two-bond system, and it maps any number of mutually non-commuting bonds to hopping on a Bethe lattice. The reason to care is that energy is the most basic conserved density, so exact control over its dynamics in a local, not all-to-all, random matrix model is a concrete step toward a hydrodynamic description of generic chaotic quantum systems.

What carries the argument

The central object is the single-particle hopping mapping. Words in the bond operators $h_j$ are read as trajectories of a particle on a graph: a fully reducible word corresponds to a returning walk, and after the large-$q$ reduction the Hamiltonian moments equal return amplitudes $\langle H^n\rangle=\langle 0|\Delta^n|0\rangle$ for a nearest-neighbor hopping operator $\Delta$. For two bonds the graph is the infinite line and the energy autocorrelator is expressed through a staggered hopping operator $V$; for $z$ mutually non-commuting bonds the same counting runs on a Bethe lattice with coordination number $z$. The proof that irreducible alternating correlators vanish uses the large-$q$ Weingarten expansion for Haar-averaged operators, with free cumulants of the symmetric Bernoulli spectrum killing all odd-cycle contributions, and the same vanishing is related to asymptotic freeness of the bond operators.

What would settle it

For the three-bond periodic configuration of Fig. 5(a), compute the empirical moments $\langle H^n\rangle$ at increasing $q$ values such as $q=12,16,20,24$ and compare them to the return amplitudes $\langle 0|\Delta^n|0\rangle$ on the $z=3$ Bethe lattice; if the normalized difference fails to decrease with $q$, or the binned eigenvalue histogram deviates from Eq. (59) beyond finite-$q$ noise, the claimed large-$q$ vanishing of irreducible words, and everything built on it, would be falsified.

Watch

Extended reading notes

Core claim

The paper establishes that the large-$q$ Haar-Ising local random matrix Hamiltonian $H=\sum_j h_j$ with $h_j^2=1$ and $\operatorname{tr}h_j=0$ behaves, for all single-trace correlation functions, like a non-interacting particle on a graph. For two overlapping bonds, the proof uses the Weingarten calculus to show $\langle(h_0h_1)^m\rangle=O(q^{-2})$, so the only words contributing to $\langle H^n\rangle$ are the fully reducible ones; these are counted by returning walks of a particle on the infinite line, giving the arcsine density of states $D(\epsilon)=\frac{1}{\pi}(4-\epsilon^2)^{-1/2}$. The same reduction lets the paper compute the staggered-hopping energy autocorrelator $C(t)=\frac{1}{2}+J_1(4t)/(4t)$, the OTOC, and the finite-temperature correlator. For $z$ mutually non-commuting terms the graph becomes the Bethe lattice, with density of states (59) and correlator (63); the paper argues the late-time decay is $t^{-3}$ for $z\ge3$ and $t^{-3/2}$ for $z=2$. In the extended $q=2$ chain, exact numerics show the autocorrelator approaches the solution of a discrete diffusion equation with diffusion constant about $0.4$, while longer-range correlators have not fully converged.

Load-bearing premise

The load-bearing premise is that in the large $q$ limit only fully reducible words of the bond operators survive, and the paper proves this in full only for two overlapping bonds; for three or more non-commuting bonds and for chains with a mix of commuting and non-commuting bonds it relies on recent results and on numerics for $L=4$ and $L=6$ chains.

Editorial extensions

If this is right

  • For two overlapping Haar-Ising bonds, the infinite-temperature energy autocorrelator decays as $t^{-3/2}$ to the equilibrium value $1/2$, with an exact Bessel-function formula, so the average system reaches equilibrium on a time scale independent of $q$.
  • The out-of-time-ordered correlator of the energy density grows as $t^2$ at early times and relaxes as $t^{-3/2}$ toward $7/8$, showing that scrambling is slower than in the all-to-all random-Hamiltonian case with energy conservation.
  • For $z\ge3$ mutually non-commuting bonds, the density of states has square-root band edges and the energy autocorrelator decays as $t^{-3}$, so adding a third non-commuting bond qualitatively changes the relaxation rate relative to the two-bond case.
  • At zero temperature the two-bond correlator $C_\beta(t)$ approaches $1$ for fixed $t$, meaning the injected energy stays localized on one bond, in contrast to the behavior argued for the GUE case.
  • In the extended $q=2$ chain, the autocorrelator matches a discrete diffusion equation with an essentially chain-length-independent diffusion constant $D\approx0.4$, but the off-diagonal correlators have not converged, so the numerics are consistent with energy diffusion without conclusively establishing it.

Reading between the lines

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

  • One testable extension the paper leaves implicit is the spectral form factor: the two-bond reducible-word counting could be adapted to two-trace quantities, and the paper expects that this requires tracking irreducible words, so a failure of the single-particle picture to reproduce the SFF ramp would locate exactly where the mapping stops.
  • The different late-time exponents ($t^{-3/2}$ for two bonds, $t^{-3}$ for $z\ge3$) suggest a clean diagnostic for whether the surrounding terms form an effectively chaotic bath, a distinction that could be probed by inserting one Haar-Ising bond into a larger non-HI random Hamiltonian.
  • If the reducible-word reduction does hold for the full chain with a mix of commuting and non-commuting bonds, the hopping-graph picture would give a constructive route to exact hydrodynamic transport coefficients in the large-$q$ limit, with the graph's geometry replacing the diffusion constant.
  • The paper's $L=4$ and $L=6$ densities of states, with their van Hove-like features, point toward a hypercube-like hopping lattice of dimension $L/2$; making that correspondence explicit would give exact finite-chain densities of states and a quantitative route to the Gaussian large-$L$ limit.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces a local random matrix Hamiltonian on a 1D lattice whose nearest-neighbor bonds are Haar-random rotations of a fixed traceless reflection (Λ²=1, tr Λ=0), dubbed the 'Haar-Ising' model. For two overlapping bonds in the large local dimension q limit, the authors argue that alternating correlators of the two bond operators vanish, which reduces the algebra of moments to counting returning walks of a single particle on a line. This yields closed-form results for the density of states (arcsine law), the infinite-temperature energy autocorrelator C(t)=1/2+J1(4t)/(4t), an OTOC, and a finite-temperature generalization. The paper extends the reduction to z mutually non-commuting bonds, mapping the dynamics to hopping on a Bethe lattice and deriving the corresponding density of states and correlator, with a t^{-3} late-time decay for z≥3. For the full 1D chain at small q, the paper presents numerical evidence of (slowly converging) diffusive energy transport, and it explores connections to free probability through numerical free convolutions.

Significance. If the central reduction is valid, the paper provides one of the few analytically solvable local random matrix models with a conserved energy. The two-term results are explicit and parameter-free, and they match exact diagonalization without fitting. The paper also makes a useful connection to free probability and gives a concrete numerical framework for the extended chain. The main caveat is that the extension to z≥3 and to chains rests on an imported large-q vanishing claim that is only weakly supported numerically.

major comments (2)
  1. [Sec. II.A, Eq. (14)] Equation (14) gives the normalized trace formula tr[(Pµ ⊗ I)(I ⊗ Pν)Pτ] = q^{-3} q^{|Orb(τµ)|} q^{|Orb(τν)|} q^{|Orb(τνµ)|}. For m=2 with µ=ν=τ=(12), this formula yields q^2, whereas a direct evaluation of the same trace over (C^q)^{⊗6} yields q^{-1}: the product operator reduces to the swap on the middle site, whose normalized trace is 1/q. The normalization in Eq. (14) should involve q^{3m} rather than q^3. Consequently, the claim in Eq. (21) that ⟨h0h1h0h1⟩ → q^{-2} appears to be incorrect; an exact evaluation of the leading large-q term gives q^{-5}. The vanishing statement of Eq. (20) remains valid, but the stated scaling and its proof need to be corrected, and the numerical check in Fig. 3(a) should be compared against the corrected exponent.
  2. [Sec. III and Appendix D] The Bethe-lattice results for z≥3 mutually non-commuting terms, and the claimed extension to the 1D chain with commuting and non-commuting bonds, rest on the assumption that all irreducible words vanish in the large q limit. The paper attributes this to Refs. [26,51] but does not provide a proof for the chain with a mixture of commuting and non-commuting bonds, and the numerical evidence in Appendix D only compares the first few total moments ⟨H^n⟩ at finite q against the reducible-word count. That test does not isolate the q-dependence of the irreducible contributions, and it does not directly probe the words a i b i that appear in the counting formula Eq. (61). Since the density of states Eq. (59), the correlator Eq. (63), and the t^{-3} late-time decay for z≥3 all depend on this assumption, the authors should either supply a proof (or a precise statement of the theorem in Refs. [26,51] and why it applies here) or provide a direct numerical scaling test that isolates the irreducible words. In the meantime, these results should be explicitly labeled as conditional.
minor comments (5)
  1. [Sec. IV, Fig. 6] The diffusion constant D=0.4 is set by eye; this is a fitted parameter, and the comparison with the diffusion equation is therefore not a parameter-free test. The authors should explicitly state this and consider showing the sensitivity of the curves to the choice of D.
  2. [General] The text contains several typos, e.g., 'corresonding' in the Fig. 3 caption and 'Kl´ ee' in the author list.
  3. [Sec. III.C] The notation 'aibi' should be typeset as a_i b_i or equivalent, to avoid confusion with the product of four operators.
  4. [Sec. V] The numerical free-convolution results are empirical and fail for the L=6 chain; this limitation should be stated in the main text rather than only in the figure caption.
  5. [Appendix D] Appendix D describes the reduction algorithm only for PBC; it would be helpful to state explicitly how the reducible-word count is obtained for OBC and for the specific correlator words used in Eq. (61).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central large-q mapping is derived in-paper via Weingarten calculus, and the analytic predictions contain no fitted parameters.

full rationale

The paper's load-bearing result, Eq. (20) for alternating correlators ⟨(h0h1)^m⟩ = O(q^-2), is derived explicitly from Weingarten calculus (Eqs. (10)–(20)) rather than assumed; the cited free-probability works [26,51] corroborate but are not the basis of the two-term proof. The consequent mapping to a single-particle walk (Eq. (30)) and the exact correlators Eq. (40), Eq. (45), and Eq. (51) are parameter-free and are benchmarked against independent exact diagonalization. The only fitted quantity, D = 0.4 in Sec. IV, sets the diffusion-equation comparison curve and is not used in any exact formula; the paper also explicitly declines to claim conclusive diffusive transport. The extension to z ≥ 3 and to full chains relies on the vanishing of irreducible words imported from Refs. [26,51], which are independent external papers rather than self-citations; the text labels this as an assumption and Appendix D supplies numerical support. Self-citations [41,44,48] appear only in background remarks on OTOCs and are not load-bearing. The acknowledged limitations, such as footnote 61 about the missing V operator and the heuristic Gaussian-moments argument in Appendix C, are openly stated rather than disguised.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central analytic results are parameter-free given the model definition, relying on standard Weingarten calculus and free probability. The main extra assumption is that irreducible correlation words vanish beyond the proven two-term case, sourced to Refs. [26,51] and checked only for short chains; this supports the Bethe-lattice mapping and the t^-3 decay. The single fitted parameter in the paper is D = 0.4 in the q = 2 transport comparison, used to test diffusion rather than to derive exact formulas.

free parameters (1)
  • Diffusion constant D = 0.4 (chosen by eye)
    In Sec. IV and Fig. 6, D is selected so the diffusion-equation solution rho0(t) matches the numerically computed autocorrelator C0(t). The paper explicitly sets it by eye and does not derive it from the Hamiltonian, and spatially distant correlators r = 1, 2 are slower to converge.
assumptions (5)
  • standard math Large-q Weingarten expansion Eq. (11): E h^{otimes m} tends to a sum over permutations weighted by q^{-2|mu|} kappa_mu(Lambda^m) P_mu.
    Taken from Ref. [52]; used throughout Sec. II to prove the vanishing of alternating correlators.
  • standard math Free cumulant structure of symmetric Bernoulli variables: odd free cumulants vanish and kappa_2 = 1 for Lambda with equal numbers of +1 and -1 eigenvalues.
    From Ref. [28]; used to show that surviving terms have even cycles and to compute the kappa values in Table I.
  • ad hoc to paper Vanishing of irreducible words beyond two terms, including z >= 3 mutually non-commuting terms and extended chains with some commuting bonds.
    Proven in-paper only for two overlapping terms. For z >= 3 and local chains the paper assumes the epsilon-free independence results of Refs. [26,51] apply, with numerical evidence only for L = 4 and L = 6 in Appendix D. The Bethe-lattice density of states and t^-3 decay depend on this.
  • domain assumption Annealed and quenched finite-temperature averages coincide in the large q limit.
    Stated in Sec. II E, citing general large-q random matrix expectations [57]; used to define the finite-temperature correlator C_beta(t) in Eq. (46).
  • standard math Bethe lattice tight-binding density of states formula Eq. (59).
    Standard result from Refs. [55,59,60]; used as the single-particle prediction for z mutually non-commuting terms.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Energy dynamics in a class of local random matrix Hamiltonians." pith.science (2026). https://pith.science/paper/MWFRFMFJ

@misc{pith2026250205045,
  author       = {Pith},
  title        = {Pith review of: Energy dynamics in a class of local random matrix Hamiltonians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MWFRFMFJ}},
  note         = {Machine review of arXiv:2502.05045}
}
read the original abstract

Random matrix theory yields valuable insights into the universal features of quantum many-body chaotic systems. Although all-to-all interactions are traditionally studied, many interesting dynamical questions, such as transport of a conserved density, require a notion of spatially local interactions. We study the transport of the energy, the most basic conserved density, in few-body and 1D chains of nearest-neighbor random matrix terms that square to one. In the few-body but large local Hilbert space dimension case, we develop a mapping for the energy dynamics to a single-particle hopping picture. This allows for the computation of the energy density autocorrelators and an out-of-time-ordered correlator of the energy density. In the 1D chain, we numerically study the energy transport for a small local Hilbert space dimension. We also discuss the density of states throughout and touch upon the relation to free probability theory.

Figures

Figures reproduced from arXiv: 2502.05045 by the authors.

Figure 1
Figure 1. FIG. 1. A local nearest-neighbor random matrix chain. Each [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Panel (a) depicts two interacting HI bonds on a three site lattice. Panel (b) is an infinite 1D lattice on which particle [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Numerical comparison of the present methods with numerics in blue and analytical theory in orange. This figure [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Various few-body scenarios of HI local terms inter [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Sketch (a) is an example of “local” random matrix with pairwise interactions that can be viewed as a 1D “chain” [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Numerical calculation of energy transport via average two-point functions [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Sketches of two small local HI random matrix Hamil [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Density of states calculated from ED for single realizations of the HI chain, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Statistics [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Numerical evidence that only reducible words con [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 53 canonical work pages

  1. [21]

    Bellitti, S

    M. Bellitti, S. Morampudi, and C. R. Laumann, Hamilto- nian dynamics of a sum of interacting random matrices, Phys. Rev. B 100, 184201 (2019)

    work page 2019

  2. [1]

    A. J. Short, Equilibration of quantum systems and sub- systems, New Journal of Physics 13, 053009 (2011)

    work page 2011

  3. [2]

    Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys

    P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101, 190403 (2008)

    work page 2008

  4. [3]

    E. P. Wigner, Random matrices in physics, SIAM review 9, 1 (1967)

    1967

  5. [4]

    Bohigas, M

    O. Bohigas, M. J. Giannoni, and C. Schmit, Character- ization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984)

    1984

  6. [5]

    A. P. Luca D’Alessio, Yariv Kafri and M. Rigol, From quantum chaos and eigenstate thermalization to statisti- cal mechanics and thermodynamics, Advances in Physics 65, 239 (2016)

    2016

  7. [6]

    Cotler, N

    J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, Chaos, complexity, and random matrices, Journal of High Energy Physics 2017, 48 (2017)

    work page 2017

  8. [7]

    A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Transition from localized to ex- tended eigenstates in the ensemble of power-law random banded matrices, Phys. Rev. E 54, 3221 (1996)

    1996

Show all 71 references
  1. [8]

    F. J. Wegner, Disordered system with n orbitals per site: n = ∞ limit, Phys. Rev. B 19, 783 (1979)

    1979

  2. [9]

    V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New Journal of Physics 17, 122002 (2015)

    2015

  3. [10]

    G. J. Rodgers and A. J. Bray, Density of states of a sparse random matrix, Phys. Rev. B 37, 3557 (1988)

    1988

  4. [11]

    Shukla, Random matrices with correlated elements: A model for disorder with interactions, Phys

    P. Shukla, Random matrices with correlated elements: A model for disorder with interactions, Phys. Rev. E 71, 026226 (2005)

    2005

  5. [12]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Ran- dom Quantum Circuits, Annual Review of Condensed Matter Physics 14, 335 (2023)

    2023

  6. [13]

    Khemani, A

    V. Khemani, A. Vishwanath, and D. A. Huse, Oper- ator Spreading and the Emergence of Dissipative Hy- drodynamics under Unitary Evolution with Conservation Laws, Physical Review X 8, 031057 (2018)

    2018

  7. [14]

    McCulloch, J

    E. McCulloch, J. De Nardis, S. Gopalakrishnan, and R. Vasseur, Full Counting Statistics of Charge in Chaotic Many-Body Quantum Systems, Physical Review Letters 131, 210402 (2023)

    2023

  8. [15]

    Von Keyserlingk, T

    C. Von Keyserlingk, T. Rakovszky, F. Pollmann, and S. Sondhi, Operator Hydrodynamics, OTOCs, and En- tanglement Growth in Systems without Conservation Laws, Physical Review X 8, 021013 (2018)

    2018

  9. [16]

    A. Chan, S. Shivam, D. A. Huse, and A. De Luca, Many- body quantum chaos and space-time translational invari- ance, Nature Communications 13, 7484 (2022)

    2022

  10. [17]

    Sachdev and J

    S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993)

    1993

  11. [18]

    Kitaev, A simple model of quantum holography,, kitp strings seminar and Entanglement 2015 program (Feb

    A. Kitaev, A simple model of quantum holography,, kitp strings seminar and Entanglement 2015 program (Feb. 12, April 7, and May 27, 2015)

    2015

  12. [19]

    Gu, X.-L

    Y. Gu, X.-L. Qi, and D. Stanford, Local criticality, diffu- sion and chaos in generalized Sachdev-Ye-Kitaev models, Journal of High Energy Physics 2017, 125 (2017)

    2017

  13. [20]

    Zanoci and B

    C. Zanoci and B. Swingle, Energy transport in Sachdev- Ye-Kitaev networks coupled to thermal baths, Physical Review Research 4, 023001 (2022)

    2022

  14. [22]

    Berkooz, M

    M. Berkooz, M. Isachenkov, V. Narovlansky, and G. Tor- rents, Towards a full solution of the large N double-scaled SYK model, Journal of High Energy Physics 2019, 79 (2019)

    2019

  15. [23]

    Verlinde, Double-scaled SYK, Chords and de Sitter Gravity (2024), arXiv:2402.00635

    H. Verlinde, Double-scaled SYK, Chords and de Sitter Gravity (2024), arXiv:2402.00635

    2024 arXiv

  16. [24]

    Erd˝ os and D

    L. Erd˝ os and D. Schr¨ oder, Phase Transition in the Den- sity of States of Quantum Spin Glasses, Mathematical Physics, Analysis and Geometry 17, 441 (2014)

    2014

  17. [25]

    S. C. Morampudi and C. R. Laumann, Many-body sys- tems with random spatially local interactions, Physical Review B 100, 245152 (2019)

    2019

  18. [26]

    Collins, Z

    B. Collins, Z. Yin, L. Zhao, and P. Zhong, The spec- trum of local random Hamiltonians, Journal of Physics A: Mathematical and Theoretical 56, 035201 (2023)

    2023

  19. [27]

    Speicher and J

    R. Speicher and J. Wysoczanski, Mixtures of classical and free independence (2016), arXiv:1603.08758

    2016 arXiv

  20. [28]

    Nica and R

    A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lec- ture Note Series (Cambridge University Press, Cam- bridge, 2006)

    2006

  21. [29]

    E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Communications in Mathemat- ical Physics 28, 251 (1972)

    1972

  22. [30]

    M. L. Mehta, Random Matrices (Elsevier, 2004)

    2004

  23. [31]

    A. Chan, A. De Luca, and J. T. Chalker, Spectral statis- tics in spatially extended chaotic quantum many-body 19 systems, Phys. Rev. Lett. 121, 060601 (2018)

    2018

  24. [32]

    Sugimoto, R

    S. Sugimoto, R. Hamazaki, and M. Ueda, Test of the eigenstate thermalization hypothesis based on local ran- dom matrix theory, Phys. Rev. Lett. 126, 120602 (2021)

    2021

  25. [33]

    Sugimoto, R

    S. Sugimoto, R. Hamazaki, and M. Ueda, Eigenstate thermalization in long-range interacting systems, Phys. Rev. Lett. 129, 030602 (2022)

    2022

  26. [34]

    Bertini, F

    B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg, and M. ˇZnidariˇ c, Finite-temperature transport in one-dimensional quantum lattice models, Rev. Mod. Phys. 93, 025003 (2021)

    2021

  27. [35]

    Gopalakrishnan and R

    S. Gopalakrishnan and R. Vasseur, Anomalous transport from hot quasiparticles in interacting spin chains, Re- ports on Progress in Physics 86, 036502 (2023)

    2023

  28. [36]

    Ljubotina, J.-Y

    M. Ljubotina, J.-Y. Desaules, M. Serbyn, and Z. Papi´ c, Superdiffusive energy transport in kinetically constrained models, Phys. Rev. X 13, 011033 (2023)

    2023

  29. [37]

    Schulz, S

    M. Schulz, S. R. Taylor, C. A. Hooley, and A. Scardicchio, Energy transport in a disordered spin chain with broken u(1) symmetry: Diffusion, subdiffusion, and many-body localization, Phys. Rev. B 98, 180201 (2018)

    2018

  30. [38]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019)

    2019

  31. [39]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics 2016, 10.1007/jhep08(2016)106 (2016)

    2016 doi

  32. [40]

    Lin and O

    C.-J. Lin and O. I. Motrunich, Out-of-time-ordered cor- relators in a quantum ising chain, Physical Review B 97, 10.1103/physrevb.97.144304 (2018)

    2018 doi

  33. [41]

    Riddell and E

    J. Riddell and E. S. Sørensen, Out-of-time ordered cor- relators and entanglement growth in the random-field xx spin chain, Phys. Rev. B 99, 054205 (2019)

    2019

  34. [42]

    Khemani, D

    V. Khemani, D. A. Huse, and A. Nahum, Velocity- dependent lyapunov exponents in many-body quantum, semiclassical, and classical chaos, Physical Review B 98, 10.1103/physrevb.98.144304 (2018)

    2018 doi

  35. [43]

    Xu and B

    S. Xu and B. Swingle, Accessing scrambling using matrix product operators, Nature Physics 16, 199–204 (2019)

    2019

  36. [44]

    Riddell, W

    J. Riddell, W. Kirkby, D. H. J. O’Dell, and E. S. Sørensen, Scaling at the out-of-time-ordered correlator wavefront: Free versus chaotic models, Phys. Rev. B108, L121108 (2023)

    2023

  37. [45]

    E. M. Fortes, I. Garc ´ ıa-Mata, R. A. Jalabert, and D. A. Wisniacki, Gauging classical and quantum integrability through out-of-time-ordered correlators, Phys. Rev. E 100, 042201 (2019)

    2019

  38. [46]

    E. M. Fortes, I. Garc ´ ıa-Mata, R. A. Jalabert, and D. A. Wisniacki, Signatures of quantum chaos transition in short spin chains, Europhysics Letters130, 60001 (2020)

    2020

  39. [47]

    Garc ´ ıa-Mata, R

    I. Garc ´ ıa-Mata, R. Jalabert, and D. Wisniacki, Out-of- time-order correlations and quantum chaos, Scholarpedia 18, 55237 (2023)

    2023

  40. [48]

    Riddell and E

    J. Riddell and E. S. Sørensen, Out-of-time-order correla- tions in the quasiperiodic aubry-andr´ e model, Phys. Rev. B 101, 024202 (2020)

    2020

  41. [49]

    Vijay and A

    S. Vijay and A. Vishwanath, Finite-temperature scram- bling of a random hamiltonian (2018), arXiv:1803.08483

    2018 arXiv

  42. [50]

    XY-model

    and the density of states of their sum is given by the free convolution, denoted a ⊞ b, where a is density of states of A and b the density of states for B in the large matrix size limit. The free convolution in some special FIG. 7. Sketches of two small local HI random matrix...

    2023

  43. [51]

    J. A. Mingo and R. Speicher, Free Probability and Ran- dom Matrices , Fields Institute Monographs, Vol. 35 (Springer, New York, NY, 2017)

    2017

  44. [52]

    Charlesworth and B

    I. Charlesworth and B. Collins, Matrix models for ε-free independence, Archiv der Mathematik 116, 585 (2021)

    2021

  45. [53]

    M. Fava, J. Kurchan, and S. Pappalardi, Designs via free probability, Phys. Rev. X 15, 011031 (2025)

    2025

  46. [54]

    Cori and G

    R. Cori and G. Hetyei, Counting genus one partitions and permutations (2013), arXiv:1306.4628 [math.CO]

    2013 arXiv

  47. [55]

    m = 4, the realization-to-realization fluctuations appear to be significantly larger (e.g

    For larger m, e.g. m = 4, the realization-to-realization fluctuations appear to be significantly larger (e.g. 80 × larger) than the average and so the scaling of the true average is difficult to see numerically

  48. [56]

    E. N. Economou, Green ’s Functions in Quantum Physics, Springer Series in Solid-State Sciences, Vol. 7 (Springer, Berlin, Heidelberg, 2006)

    2006

  49. [57]

    By using that ∆ is translation invariant, we have ⟨0|V (t)V (0)|0⟩ = ⟨x|(TxV T† x )(t)(TxV T† x )(0)|x⟩

    This is not exactly trivial; but follows by inserting I = T † x Tx where Tx translates by x sites: ⟨0|V (t)V (0)|0⟩ = ⟨0|T † x TxV (t)T † x TxV (0)T † x Tx|0⟩ . By using that ∆ is translation invariant, we have ⟨0|V (t)V (0)|0⟩ = ⟨x|(TxV T† x )(t)(TxV T† x )(0)|x⟩ . If x is ev...

  50. [58]

    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, Journal of High Energy Physics 2017, 118 (2017)

    2017

  51. [59]

    G. D. Mahan, Energy bands of the Bethe lattice, Physical Review B 63, 155110 (2001)

    2001

  52. [60]

    Giacometti, Exact closed form of the return proba- bility on the bethe lattice, Journal of Physics A: Mathe- matical and General 28, L13 (1995)

    A. Giacometti, Exact closed form of the return proba- bility on the bethe lattice, Journal of Physics A: Mathe- matical and General 28, L13 (1995)

    1995

  53. [61]

    Eckstein, M

    M. Eckstein, M. Kollar, K. Byczuk, and D. Vollhardt, Hopping on the Bethe lattice: Exact results for densi- ties of states and dynamical mean-field theory, Physical Review B 71, 235119 (2005)

    2005

  54. [62]

    We were not able to identify an operator V such that C(t) = ⟨0|V (t)V (0)|0⟩ as we were in the z = 2 case, the main issue being that on the Bethe lattice (z >2), points are not labeled by d anymore

  55. [63]

    Skilling, ed., Maximum Entropy and Bayesian Meth- ods: Cambridge, England, 1988 (Springer Netherlands, Dordrecht, 1989)

    J. Skilling, ed., Maximum Entropy and Bayesian Meth- ods: Cambridge, England, 1988 (Springer Netherlands, Dordrecht, 1989)

    1988

  56. [64]

    T. J. Park and J. C. Light, Unitary quantum time evolu- tion by iterative lanczos reduction, The Journal of Chem- ical Physics 85, 5870 (1986)

    1986

  57. [65]

    M lotkowski, λ-free probability, Infinite Dimensional Analysis, Quantum Probability and Related Topics 07, 27 (2004), https://doi.org/10.1142/S0219025704001517

    W. M lotkowski, λ-free probability, Infinite Dimensional Analysis, Quantum Probability and Related Topics 07, 27 (2004), https://doi.org/10.1142/S0219025704001517

    2004 doi

  58. [66]

    C´ ebron, P

    G. C´ ebron, P. O. Santos, and P. Youssef, Graphon- Theoretic Approach to Central Limit Theorems for ε- Independence (2024), arXiv:2411.13062

    2024 arXiv

  59. [67]

    J. P. Keating, N. Linden, and H. J. Wells, Spectra and Eigenstates of Spin Chain Hamiltonians, Communica- tions in Mathematical Physics 338, 81 (2015)

    2015

  60. [68]

    Speicher, Free probability lecture notes (2019), lecture notes from Saarland University

    R. Speicher, Free probability lecture notes (2019), lecture notes from Saarland University

    2019

  61. [69]

    Novak and M

    J. Novak and M. LaCroix, Three lectures on free proba- bility (2012), arXiv:1205.2097

    2012 arXiv

  62. [70]

    Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013). 20

    2013

  63. [71]

    We thank John Chalker for pointing out the beginnings of this argument to us

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.