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arxiv: 2509.15295 · v2 · submitted 2025-09-18 · ✦ hep-th · gr-qc· quant-ph

Living on the edge: a non-perturbative resolution to the negativity of bulk entropies

Stefano Antonini , Luca V. Iliesiu , Pratik Rath , Patrick Tran This is my paper

Pith reviewed 2026-05-18 15:44 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords black hole entropyentanglement negativitymatrix integralnon-perturbative effectsRényi entropytwo-sided black holesrandom tensor networks
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The pith

Non-perturbative saddles in the dual matrix integral restore positivity to the entanglement entropies of two-sided black holes with many interior excitations.

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

The paper addresses an information paradox in which the entanglement and Rényi entropies computed from the gravitational path integral for two-sided black holes turn negative once a large number of matter excitations sit behind the horizon. This negativity cannot occur in ordinary quantum mechanics, so the authors sum over all non-perturbative contributions rather than stopping at the genus expansion. In the dual matrix-integral description that extends beyond the perturbative regime, positivity is restored by the dominance of a one-eigenvalue instanton and a two-eigenvalue instanton. The same resolution appears when the problem is recast in random tensor-network models. A sympathetic reader cares because a consistent quantum description of black holes requires that entropy quantities remain non-negative.

Core claim

The central claim is that the negativity of bulk entanglement and Rényi entropies, first identified by LMRS for BPS two-sided black holes and shown to persist more generally, is eliminated once every non-perturbative contribution to the gravitational path integral is included; in the regime where the dual matrix integral is reliable, the leading such contributions are a one-eigenvalue instanton and a two-eigenvalue instanton whose saddles enforce positive values even for arbitrarily large numbers of matter excitations behind the horizon.

What carries the argument

The one-eigenvalue instanton and two-eigenvalue instanton saddles of the dual matrix integral, which supply the non-perturbative corrections that dominate beyond genus re-summation and enforce non-negative entropies.

If this is right

  • The resolution holds for both supersymmetric BPS black holes and for non-supersymmetric two-sided black holes.
  • Random tensor-network models exhibit an analogous negativity puzzle that is cured by the same non-perturbative mechanism.
  • Entanglement and Rényi entropies remain strictly non-negative for any number of matter excitations once the instanton saddles are summed.
  • The matrix-integral description supplies a controlled way to compute entropies beyond the perturbative genus expansion of the gravitational path integral.

Where Pith is reading between the lines

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

  • Similar instanton effects may resolve other instances of apparent negativity or information loss in holographic calculations.
  • The result suggests that the spectrum of the dual CFT must contain additional non-perturbative states whose contributions cancel the would-be negative entropies.
  • The technique could be tested by constructing explicit matrix models for higher-genus or higher-dimensional black-hole geometries and checking whether the same instanton saddles appear.

Load-bearing premise

The dual matrix integral continues to capture the full gravitational path integral outside the genus-re-summation regime and the newly identified one- and two-eigenvalue instantons are the dominant contributions that restore positivity.

What would settle it

An explicit evaluation of the Rényi entropy that includes all higher-order non-perturbative effects and finds a negative value for some finite but large number of matter excitations would falsify the proposed resolution.

read the original abstract

Lin, Maldacena, Rozenberg, and Shan (LMRS) presented a new information paradox in black hole physics by noticing that the entanglement and R\'enyi entropies in a two-sided black hole can become negative when the geometry contains a very large number of matter excitations behind the black hole horizon. While originally this puzzle was presented in the context of BPS two-sided black holes in two-dimensional supergravity, the negativity in fact persists for more general two-sided black holes in the presence of a large number of matter excitations. Since the entanglement and R\'enyi entropies in ordinary quantum systems cannot be negative, resolving this puzzle is a necessary step towards understanding the quantum mechanical description of black holes. In this paper, we explain how to address the entanglement negativity puzzle, both in the original setting discussed by LMRS and in more general non-supersymmetric settings, by summing over all non-perturbative contributions to the gravitational path integral. We then interpret this result from the point of view of a dual matrix integral, which we use to extend our analysis beyond the regime of validity of the genus re-summation performed in the gravitational path integral. In this regime, positivity is rescued by new saddles of the matrix integral, a one-eigenvalue instanton and a two-eigenvalue instanton. Finally, we formulate a similar puzzle and its resolution using random tensor network techniques.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper addresses the negativity of entanglement and Rényi entropies for two-sided black holes containing a large number of matter excitations, as first noted by LMRS. It proposes a resolution by summing all non-perturbative contributions to the gravitational path integral. The analysis is extended via a dual matrix integral beyond the regime of genus resummation, where positivity is restored by the contributions of a one-eigenvalue instanton and a two-eigenvalue instanton. A parallel formulation and resolution of the puzzle is given using random tensor networks.

Significance. If the dominance and sign-flipping effect of the identified instantons can be established with explicit calculations, the result would supply a concrete non-perturbative mechanism ensuring non-negative entropies in a gravitational setting. The dual matrix-integral and tensor-network perspectives offer useful cross-checks on the gravitational path integral and extend the analysis past perturbative resummation, which would be a substantive advance for the black-hole information problem if substantiated.

major comments (2)
  1. [dual matrix integral section] Section on the dual matrix integral: the assertion that the one- and two-eigenvalue instantons dominate and restore positivity for large matter excitations is stated without an explicit scaling analysis, action computation, or numerical check showing that their contribution overcomes the negative genus-resummed term.
  2. [matrix integral extension] The mapping from the gravitational path integral to the matrix integral is used to access the non-perturbative regime, yet no error estimate or matching condition is supplied to confirm that the newly identified saddles remain the leading contributions once the number of matter excitations becomes large.
minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief recap of the precise entropy expression (from LMRS) that becomes negative, to make the starting point of the puzzle fully explicit.
  2. [matrix integral section] Notation for the instanton actions and their relation to the resolvent or replica computation could be collected in a short table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the non-perturbative analysis. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [dual matrix integral section] Section on the dual matrix integral: the assertion that the one- and two-eigenvalue instantons dominate and restore positivity for large matter excitations is stated without an explicit scaling analysis, action computation, or numerical check showing that their contribution overcomes the negative genus-resummed term.

    Authors: We agree that an explicit scaling analysis and action computation would strengthen the presentation. In the revised version we will add a dedicated subsection that computes the on-shell actions of the one- and two-eigenvalue instantons, derives their scaling with the number of matter excitations, and shows analytically that these contributions become parametrically larger than the negative genus-resummed term in the regime of interest. We will also outline how a numerical check could be performed in a truncated matrix model. revision: yes

  2. Referee: [matrix integral extension] The mapping from the gravitational path integral to the matrix integral is used to access the non-perturbative regime, yet no error estimate or matching condition is supplied to confirm that the newly identified saddles remain the leading contributions once the number of matter excitations becomes large.

    Authors: The referee correctly notes the absence of a quantitative error estimate. We will revise the text to include a discussion of the matching conditions between the gravitational path integral and the matrix integral, together with an estimate of the sub-leading corrections that arise when the number of matter excitations is taken large. This addition will clarify the regime in which the one- and two-eigenvalue saddles dominate. revision: yes

Circularity Check

0 steps flagged

No significant circularity: resolution via new matrix integral saddles is independent of input negativity

full rationale

The paper's derivation proceeds by first reproducing the negativity from the genus-resummed gravitational path integral (as in LMRS), then extending the analysis by summing non-perturbative contributions and mapping to a dual matrix integral whose new one- and two-eigenvalue saddles are shown to restore positivity. This extension is external to the original perturbative calculation and does not reduce any claimed result to a fitted parameter, self-definition, or self-citation chain. The central steps rely on explicit saddle identification and the matrix integral's non-perturbative structure rather than re-expressing the input negativity by construction. The derivation is therefore self-contained against the external benchmark of the matrix model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the existence and dominance of non-perturbative contributions and new instanton saddles whose independent evidence is not supplied outside the calculation itself.

axioms (1)
  • domain assumption The gravitational path integral can be summed over all non-perturbative contributions to produce a well-defined entropy.
    Invoked to resolve negativity in both supersymmetric and non-supersymmetric settings.
invented entities (2)
  • one-eigenvalue instanton no independent evidence
    purpose: New saddle that rescues positivity in the matrix-integral regime.
    Introduced as the dominant contribution beyond genus re-summation.
  • two-eigenvalue instanton no independent evidence
    purpose: New saddle that rescues positivity in the matrix-integral regime.
    Introduced as the dominant contribution beyond genus re-summation.

pith-pipeline@v0.9.0 · 5801 in / 1389 out tokens · 44530 ms · 2026-05-18T15:44:08.224815+00:00 · methodology

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Forward citations

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  2. Exploring the Spectral Edge in SYK Models

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    Numerical confirmation that SYK models reproduce RMT spectral edge statistics, yielding power-law quenched entropy at low T and enabling large-N entanglement entropy calculations for supersymmetric wormholes.

Reference graph

Works this paper leans on

134 extracted references · 134 canonical work pages · cited by 2 Pith papers · 27 internal anchors

  1. [1]

    Entanglement Wedge Reconstruction and the Information Paradox

    G. Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  2. [2]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield,The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,JHEP12(2019) 063 [1905.08762]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  3. [3]

    Penington, S.H

    G. Penington, S.H. Shenker, D. Stanford and Z. Yang,Replica wormholes and the black hole interior,JHEP03(2022) 205 [1911.11977]

    work page arXiv 2022

  4. [4]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,Replica Wormholes and the Entropy of Hawking Radiation,JHEP05(2020) 013 [1911.12333]

    work page arXiv 2020

  5. [5]

    JT gravity as a matrix integral

    P. Saad, S.H. Shenker and D. Stanford,JT gravity as a matrix integral,1903.11115

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  6. [6]

    Saad,Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 1910.10311

    P. Saad,Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 1910.10311

    work page arXiv 1910

  7. [7]

    Iliesiu, S

    L.V. Iliesiu, S. Murthy and G.J. Turiaci,Black hole microstate counting from the gravitational path integral,2209.13602

    work page arXiv

  8. [8]

    H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan,Looking at supersymmetric black holes for a very long time,SciPost Phys.14(2023) 128 [2207.00408]

    work page arXiv 2023

  9. [9]

    H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan,Holography for people with no time, SciPost Phys.14(2023) 150 [2207.00407]

    work page arXiv 2023

  10. [10]

    Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK

    A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde,Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK,JHEP02(2019) 156 [1807.03916]. – 96 –

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  11. [11]

    Hsin, L.V

    P.-S. Hsin, L.V. Iliesiu and Z. Yang,A violation of global symmetries from replica wormholes and the fate of black hole remnants,Class. Quant. Grav.38(2021) 194004 [2011.09444]

    work page arXiv 2021

  12. [12]

    Sasieta,Wormholes from heavy operator statistics in AdS/CFT,JHEP03(2023) 158 [2211.11794]

    M. Sasieta,Wormholes from heavy operator statistics in AdS/CFT,JHEP03(2023) 158 [2211.11794]

    work page arXiv 2023

  13. [13]

    Balasubramanian, A

    V. Balasubramanian, A. Lawrence, J.M. Magan and M. Sasieta,Microscopic Origin of the Entropy of Black Holes in General Relativity,Phys. Rev. X14(2024) 011024 [2212.02447]

    work page arXiv 2024

  14. [14]

    Balasubramanian, A

    V. Balasubramanian, A. Lawrence, J.M. Magan and M. Sasieta,Microscopic Origin of the Entropy of Astrophysical Black Holes,Phys. Rev. Lett.132(2024) 141501 [2212.08623]

    work page arXiv 2024

  15. [15]

    Antonini, M

    S. Antonini, M. Sasieta and B. Swingle,Cosmology from random entanglement,JHEP11 (2023) 188 [2307.14416]

    work page arXiv 2023

  16. [16]

    de Boer, D

    J. de Boer, D. Liska, B. Post and M. Sasieta,A principle of maximum ignorance for semiclassical gravity,JHEP2024(2024) 003 [2311.08132]

    work page arXiv 2024

  17. [17]

    Antonini and P

    S. Antonini and P. Rath,Do holographic CFT states have unique semiclassical bulk duals?, 2408.02720

    work page arXiv

  18. [18]

    Magan, M

    J.M. Magan, M. Sasieta and B. Swingle,ER for typical EPR,2504.07171

    work page arXiv

  19. [19]

    Antonini, P

    S. Antonini, P. Rath, M. Sasieta, B. Swingle and A. Vilar L´ opez,The Baby Universe is Fine and the CFT Knows It: On Holography for Closed Universes,2507.10649

    work page arXiv

  20. [20]

    Engelhardt and E

    N. Engelhardt and E. Gesteau,Further Evidence Against a Semiclassical Baby Universe in AdS/CFT,2504.14586

    work page arXiv

  21. [21]

    Engelhardt, S

    N. Engelhardt, S. Fischetti and A. Maloney,Free energy from replica wormholes,Phys. Rev. D103(2021) 046021 [2007.07444]

    work page arXiv 2021

  22. [22]

    Hern´ andez-Cuenca,Entropy and spectrum of near-extremal black holes: semiclassical brane solutions to non-perturbative problems,JHEP05(2025) 020 [2407.20321]

    S. Hern´ andez-Cuenca,Entropy and spectrum of near-extremal black holes: semiclassical brane solutions to non-perturbative problems,JHEP05(2025) 020 [2407.20321]

    work page arXiv 2025

  23. [23]

    Antonini, L.V

    S. Antonini, L.V. Iliesiu, P. Rath and P. Tran,A Black Hole Airy Tail,2507.10657

    work page arXiv

  24. [24]

    Boruch, L.V

    J. Boruch, L.V. Iliesiu and C. Yan,Constructing all BPS black hole microstates from the gravitational path integral,JHEP09(2024) 058 [2307.13051]

    work page arXiv 2024

  25. [25]

    Boruch, M.T

    J. Boruch, M.T. Heydeman, L.V. Iliesiu and G.J. Turiaci,BPS and near-BPS black holes in AdS5 and their spectrum inN= 4 SYM,JHEP07(2025) 220 [2203.01331]

    work page arXiv 2025

  26. [26]

    Heydeman, L.V

    M. Heydeman, L.V. Iliesiu, G.J. Turiaci and W. Zhao,The statistical mechanics of near-BPS black holes,J. Phys. A55(2022) 014004 [2011.01953]

    work page arXiv 2022

  27. [27]

    Lin, L.V

    G. Lin, L.V. Iliesiu and M. Usatyuk,The evaporation of black holes in supergravity,JHEP 08(2025) 220 [2504.21077]

    work page arXiv 2025

  28. [28]

    Fermionic Localization of the Schwarzian Theory

    D. Stanford and E. Witten,Fermionic Localization of the Schwarzian Theory,JHEP10 (2017) 008 [1703.04612]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  29. [29]

    Penington and E

    G. Penington and E. Witten,Algebras and states in super-JT gravity,2412.15549

    work page arXiv

  30. [30]

    A tauberian theorem for the conformal bootstrap

    J. Qiao and S. Rychkov,A tauberian theorem for the conformal bootstrap,JHEP12(2017) 119 [1709.00008]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  31. [31]

    Holographic duality from random tensor networks

    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang,Holographic duality from random tensor networks,JHEP11(2016) 009 [1601.01694]. – 97 –

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  32. [32]

    Antonini, G

    S. Antonini, G. Bentsen, C. Cao, J. Harper, S.-K. Jian and B. Swingle,Holographic measurement and bulk teleportation,JHEP12(2022) 124 [2209.12903]

    work page arXiv 2022

  33. [33]

    Solving the Schwarzian via the Conformal Bootstrap

    T.G. Mertens, G.J. Turiaci and H.L. Verlinde,Solving the Schwarzian via the Conformal Bootstrap,JHEP08(2017) 136 [1705.08408]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [34]

    The Schwarzian Theory - A Wilson Line Perspective

    A. Blommaert, T.G. Mertens and H. Verschelde,The Schwarzian Theory - A Wilson Line Perspective,JHEP12(2018) 022 [1806.07765]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  35. [35]

    Iliesiu, S.S

    L.V. Iliesiu, S.S. Pufu, H. Verlinde and Y. Wang,An exact quantization of Jackiw-Teitelboim gravity,JHEP11(2019) 091 [1905.02726]

    work page arXiv 2019

  36. [36]

    Yang,The Quantum Gravity Dynamics of Near Extremal Black Holes,JHEP05(2019) 205 [1809.08647]

    Z. Yang,The Quantum Gravity Dynamics of Near Extremal Black Holes,JHEP05(2019) 205 [1809.08647]

    work page arXiv 2019

  37. [37]

    Iliesiu, M

    L.V. Iliesiu, M. Kologlu and G.J. Turiaci,Supersymmetric indices factorize,JHEP05 (2023) 032 [2107.09062]

    work page arXiv 2023

  38. [38]

    Boruch, L.V

    J. Boruch, L.V. Iliesiu, G. Lin and C. Yan,How the Hilbert space of two-sided black holes factorises,JHEP06(2025) 092 [2406.04396]

    work page arXiv 2025

  39. [39]

    Blommaert,Dissecting the ensemble in JT gravity,JHEP09(2022) 075 [2006.13971]

    A. Blommaert,Dissecting the ensemble in JT gravity,JHEP09(2022) 075 [2006.13971]

    work page arXiv 2022

  40. [40]

    Iliesiu, M

    L.V. Iliesiu, M. Mezei and G. S´ arosi,The volume of the black hole interior at late times, JHEP07(2022) 073 [2107.06286]

    work page arXiv 2022

  41. [41]

    Walsh and A

    T. Walsh and A. Lehman,Counting rooted maps by genus. i,Journal of Combinatorial Theory, Series B13(1972) 192

    work page 1972

  42. [42]

    Walsh and A

    T. Walsh and A. Lehman,Counting rooted maps by genus ii,Journal of Combinatorial Theory, Series B13(1972) 122

    work page 1972

  43. [43]

    Zagier and J

    D. Zagier and J. Harer,The euler characteristic of the moduli space of curves.,Inventiones mathematicae85(1986) 457

    work page 1986

  44. [44]

    Jackson,Counting cycles in permutations by group characters, with an application to a topological problem,Transactions of the American Mathematical Society299(1987) 785

    D.M. Jackson,Counting cycles in permutations by group characters, with an application to a topological problem,Transactions of the American Mathematical Society299(1987) 785

    work page 1987

  45. [45]

    Coquereaux and J.-B

    R. Coquereaux and J.-B. Zuber,Counting partitions by genus: a compendium of results, arXiv e-prints(2023) arXiv:2305.01100 [2305.01100]

    work page arXiv 2023

  46. [46]

    Generating series for GUE correlators

    B. Dubrovin and D. Yang,Generating series for GUE correlators,Letters in Mathematical Physics107(2017) 1971 [1604.07628]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  47. [47]

    Towards a full solution of the large N double-scaled SYK model

    M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents,Towards a full solution of the large N double-scaled SYK model,JHEP03(2019) 079 [1811.02584]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  48. [48]

    Lin,The bulk Hilbert space of double scaled SYK,JHEP11(2022) 060 [2208.07032]

    H.W. Lin,The bulk Hilbert space of double scaled SYK,JHEP11(2022) 060 [2208.07032]

    work page arXiv 2022

  49. [49]

    Marolf and H

    D. Marolf and H. Maxfield,Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,JHEP08(2020) 044 [2002.08950]

    work page arXiv 2020

  50. [50]

    Belin and J

    A. Belin and J. de Boer,Random statistics of OPE coefficients and Euclidean wormholes, Class. Quant. Grav.38(2021) 164001 [2006.05499]

    work page arXiv 2021

  51. [51]

    Saad, S.H

    P. Saad, S.H. Shenker and S. Yao,Comments on wormholes and factorization,JHEP10 (2024) 076 [2107.13130]

    work page arXiv 2024

  52. [52]

    Chandra, S

    J. Chandra, S. Collier, T. Hartman and A. Maloney,Semiclassical 3D gravity as an average of large-c CFTs,JHEP12(2022) 069 [2203.06511]. – 98 –

    work page arXiv 2022

  53. [53]

    Chandra and T

    J. Chandra and T. Hartman,Coarse graining pure states in AdS/CFT,JHEP10(2023) 030 [2206.03414]

    work page arXiv 2023

  54. [54]

    Wormholes in AdS

    J.M. Maldacena and L. Maoz,Wormholes in AdS,JHEP02(2004) 053 [hep-th/0401024]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  55. [55]

    The Large N Limit of Superconformal Field Theories and Supergravity

    J.M. Maldacena,The Large N limit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  56. [56]

    Large N Field Theories, String Theory and Gravity

    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz,Large N field theories, string theory and gravity,Phys. Rept.323(2000) 183 [hep-th/9905111]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  57. [57]

    N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals

    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena,N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,JHEP10(2008) 091 [0806.1218]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  58. [58]

    Mukhametzhanov,Factorization and complex couplings in SYK and in Matrix Models, JHEP04(2023) 122 [2110.06221]

    B. Mukhametzhanov,Factorization and complex couplings in SYK and in Matrix Models, JHEP04(2023) 122 [2110.06221]

    work page arXiv 2023

  59. [59]

    Blommaert, L.V

    A. Blommaert, L.V. Iliesiu and J. Kruthoff,Gravity factorized,JHEP09(2022) 080 [2111.07863]

    work page arXiv 2022

  60. [60]

    Saad, S.H

    P. Saad, S.H. Shenker, D. Stanford and S. Yao,Wormholes without averaging,JHEP09 (2024) 133 [2103.16754]

    work page arXiv 2024

  61. [61]

    Blommaert, L.V

    A. Blommaert, L.V. Iliesiu and J. Kruthoff,Alpha states demystified — towards microscopic models of AdS2 holography,JHEP08(2022) 071 [2203.07384]

    work page arXiv 2022

  62. [62]

    Chaos and Quantum Thermalization

    M. Srednicki,Chaos and Quantum Thermalization,Phys. Rev. E50(1994) [cond-mat/9403051]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  63. [63]

    Deutsch,Quantum statistical mechanics in a closed system,Physical review a43 (1991) 2046

    J.M. Deutsch,Quantum statistical mechanics in a closed system,Physical review a43 (1991) 2046

    work page 1991

  64. [64]

    Jafferis, D.K

    D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner,Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices, Phys. Rev. D108(2023) 066015 [2209.02131]

    work page arXiv 2023

  65. [65]

    Jafferis, D.K

    D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner,Matrix Models for Eigenstate Thermalization,Phys. Rev. X13(2023) 031033 [2209.02130]

    work page arXiv 2023

  66. [66]

    Anninos and B

    D. Anninos and B. M¨ uhlmann,Notes on matrix models (matrix musings),J. Stat. Mech. 2008(2020) 083109 [2004.01171]

    work page arXiv 2008

  67. [67]

    Wigner,Characteristic vectors of bordered matrices with infinite dimensions ii, inThe Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, A.S

    E.P. Wigner,Characteristic vectors of bordered matrices with infinite dimensions ii, inThe Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, A.S. Wightman, ed., (Berlin, Heidelberg), pp. 541–545, Springer Berlin Heidelberg (1993), DOI

    work page 1993

  68. [68]

    Dyson,Statistical theory of the energy levels of complex systems

    F.J. Dyson,Statistical theory of the energy levels of complex systems. i,Journal of Mathematical Physics3(1962) 140 [https://pubs.aip.org/aip/jmp/article-pdf/3/1/140/19079507/140 1 online.pdf]

    work page 1962

  69. [69]

    Dyson,The threefold way

    F.J. Dyson,The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics,Journal of Mathematical Physics3(1962) 1199 [https://pubs.aip.org/aip/jmp/article-pdf/3/6/1199/19064331/1199 1 online.pdf]

    work page 1962

  70. [70]

    Tracy and H

    C.A. Tracy and H. Widom,Level-spacing distributions and the airy kernel,Physics Letters B305(1993) 115–118

    work page 1993

  71. [71]

    Brezin and S

    E. Brezin and S. Hikami,Vertices from replica in a random matrix theory,Journal of Physics A: Mathematical and Theoretical40(2007) 13545–13566. – 99 –

    work page 2007

  72. [72]

    Br´ ezin and S

    E. Br´ ezin and S. Hikami,Random Matrix Theory with an External Source, vol. 19 of SpringerBriefs in Mathematical Physics, Springer (2016), 10.1007/978-981-10-3316-2

    work page doi:10.1007/978-981-10-3316-2 2016

  73. [73]

    Bornemann,Asymptotic independence of the extreme eigenvalues of gaussian unitary ensemble,Journal of Mathematical Physics51(2010)

    F. Bornemann,Asymptotic independence of the extreme eigenvalues of gaussian unitary ensemble,Journal of Mathematical Physics51(2010)

    work page 2010

  74. [74]

    Kontsevich,Intersection theory on the moduli space of curves and the matrix Airy function,Commun

    M. Kontsevich,Intersection theory on the moduli space of curves and the matrix Airy function,Commun. Math. Phys.147(1992) 1

    work page 1992

  75. [75]

    Janssen and M

    O. Janssen and M. Mirbabayi,Low-temperature entropy in JT gravity,JHEP06(2021) 074 [2103.03896]

    work page arXiv 2021

  76. [76]

    Ginibre,Statistical Ensembles of Complex, Quaternion and Real Matrices,J

    J. Ginibre,Statistical Ensembles of Complex, Quaternion and Real Matrices,J. Math. Phys. 6(1965) 440

    work page 1965

  77. [77]

    Entanglement Renormalization and Holography

    B. Swingle,Entanglement Renormalization and Holography,Phys. Rev. D86(2012) 065007 [0905.1317]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  78. [78]

    Bulk Locality and Quantum Error Correction in AdS/CFT

    A. Almheiri, X. Dong and D. Harlow,Bulk Locality and Quantum Error Correction in AdS/CFT,JHEP04(2015) 163 [1411.7041]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  79. [79]

    Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill,Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence,JHEP06(2015) 149 [1503.06237]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  80. [80]

    Holographic Renyi Entropy from Quantum Error Correction

    C. Akers and P. Rath,Holographic Renyi Entropy from Quantum Error Correction,JHEP 05(2019) 052 [1811.05171]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

Showing first 80 references.