@misc{Wang2026OptimalMixing,
  author = {Wang, Sihan},
  title = {Optimal Mixing of {Glauber} Dynamics for the {Sherrington--Kirkpatrick} Model at $\beta<1/2$},
  year = {2026},
  eprint = {2608.22159},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  note = {Version 2, August 28, 2026},
  url = {https://arxiv.org/abs/2608.22159v2}
}

@misc{BobanLiOveisGharan2026,
  author = {Boban, Mathews and Li, Anqi and Oveis Gharan, Shayan},
  title = {Rank-1-perturbed trickledown theorems: Mixing time of {Glauber} dynamics for the {Sherrington--Kirkpatrick} model up to $\beta\leq\frac{1}{2}+\varepsilon$},
  year = {2026},
  eprint = {2609.13138},
  archivePrefix = {arXiv},
  primaryClass = {cs.DS},
  note = {Version 1, September 11, 2026},
  url = {https://arxiv.org/abs/2609.13138v1}
}

@article{EldanKoehlerZeitouni2022,
  author = {Eldan, Ronen and Koehler, Frederic and Zeitouni, Ofer},
  title = {A spectral condition for spectral gap: fast mixing in high-temperature {Ising} models},
  journal = {Probability Theory and Related Fields},
  volume = {182}, number = {3--4}, pages = {1035--1051}, year = {2022},
  doi = {10.1007/s00440-021-01085-x},
  eprint = {2007.08200}, archivePrefix = {arXiv},
  note = {Preprint version 2, August 6, 2021},
  url = {https://arxiv.org/abs/2007.08200v2}
}

@inproceedings{AnariKoehlerVuong2024,
  author = {Anari, Nima and Koehler, Frederic and Vuong, Thuy-Duong},
  title = {Trickle-Down in Localization Schemes and Applications},
  booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing},
  pages = {1094--1105}, year = {2024},
  doi = {10.1145/3618260.3649622},
  eprint = {2407.16104}, archivePrefix = {arXiv},
  note = {Full preprint version 1, July 23, 2024},
  url = {https://arxiv.org/abs/2407.16104v1}
}

@article{BilloireCampbell2011CriticalDynamics,
  author = {Billoire, Alain and Campbell, I. A.},
  title = {Dynamics in the {Sherrington--Kirkpatrick Ising} spin glass at and above {$T_g$}},
  journal = {Physical Review B},
  volume = {84},
  pages = {054442},
  year = {2011},
  doi = {10.1103/PhysRevB.84.054442},
  eprint = {1105.1902},
  archivePrefix = {arXiv},
  note = {arXiv version 3, 28 July 2011},
  url = {https://arxiv.org/abs/1105.1902v3}
}

@misc{DuHuang2026CriticalFluctuations,
  author = {Du, Hang and Huang, Brice},
  title = {Fluctuations of the {Sherrington--Kirkpatrick} free energy at critical temperature},
  year = {2026},
  eprint = {2607.02172},
  archivePrefix = {arXiv},
  note = {Version 2, 8 July 2026},
  url = {https://arxiv.org/abs/2607.02172v2}
}

@misc{DuHuang2026CriticalOverlap,
  author = {Du, Hang and Huang, Brice},
  title = {Overlap distribution of the critical {Sherrington--Kirkpatrick} model},
  year = {2026},
  eprint = {2608.08752},
  archivePrefix = {arXiv},
  note = {Version 1, 9 August 2026},
  url = {https://arxiv.org/abs/2608.08752v1}
}

@article{GMT2006,
  author = {Goel, Sharad and Montenegro, Ravi and Tetali, Prasad},
  title = {Mixing Time Bounds via the Spectral Profile},
  journal = {Electronic Journal of Probability},
  volume = {11},
  number = {1},
  pages = {1--26},
  year = {2006},
  doi = {10.1214/EJP.v11-300},
  url = {https://emis.de/ft/42421},
  eprint = {math/0505690},
  archivePrefix = {arXiv}
}

@article{DSC1996,
  author = {Diaconis, Persi and Saloff-Coste, Laurent},
  title = {Logarithmic {Sobolev} inequalities for finite {Markov} chains},
  journal = {The Annals of Applied Probability},
  volume = {6},
  number = {3},
  pages = {695--750},
  year = {1996},
  doi = {10.1214/aoap/1034968224},
  url = {https://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/ConvergeRates/DiaconisSaloffCosteSobolev1996.pdf}
}

@article{BR2003,
  author = {Barthe, Franck and Roberto, Cyril},
  title = {{Sobolev} inequalities for probability measures on the real line},
  journal = {Studia Mathematica},
  volume = {159},
  number = {3},
  pages = {481--497},
  year = {2003},
  doi = {10.4064/sm159-3-9},
  url = {https://www.impan.pl/shop/publication/transaction/download/product/89465}
}

@article{BobkovTetali2006Modified,
  author = {Bobkov, Sergey G. and Tetali, Prasad},
  title = {Modified Logarithmic {Sobolev} Inequalities in Discrete Settings},
  journal = {Journal of Theoretical Probability},
  volume = {19},
  number = {2},
  pages = {289--336},
  year = {2006},
  doi = {10.1007/s10959-006-0016-3},
  url = {https://www-users.cse.umn.edu/~bobko001/papers/2006_JTP_BT_Modified.pdf}
}

@article{Comets1996Spherical,
  author = {Comets, Francis},
  title = {A spherical bound for the {Sherrington--Kirkpatrick} model},
  journal = {Ast{\'e}risque},
  volume = {236},
  pages = {103--108},
  year = {1996},
  url = {https://www.numdam.org/item/AST_1996__236__103_0/},
  note = {In Hommage {\`a} P. A. Meyer et J. Neveu}
}

@article{DumitriuEdelman2002,
  author = {Dumitriu, Ioana and Edelman, Alan},
  title = {Matrix models for beta ensembles},
  journal = {Journal of Mathematical Physics},
  volume = {43},
  number = {11},
  pages = {5830--5847},
  year = {2002},
  doi = {10.1063/1.1507823},
  eprint = {math-ph/0206043},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/math-ph/0206043v1}
}

@article{LedouxRider2010,
  author = {Ledoux, Michel and Rider, Brian},
  title = {Small deviations for beta ensembles},
  journal = {Electronic Journal of Probability},
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  number = {41},
  pages = {1319--1343},
  year = {2010},
  doi = {10.1214/EJP.v15-798},
  eprint = {0912.5040},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/0912.5040v1}
}

@article{RamirezRiderVirag2011,
  author = {Ram{\'i}rez, Jos{\'e} A. and Rider, Brian and Vir{\'a}g, B{\'a}lint},
  title = {Beta ensembles, stochastic {Airy} spectrum, and a diffusion},
  journal = {Journal of the American Mathematical Society},
  volume = {24},
  number = {4},
  pages = {919--944},
  year = {2011},
  doi = {10.1090/S0894-0347-2011-00703-0},
  eprint = {math/0607331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/math/0607331v4}
}

@article{BrascampLieb1976,
  author = {Brascamp, Herm Jan and Lieb, Elliott H.},
  title = {On extensions of the {Brunn--Minkowski} and {Pr{\'e}kopa--Leindler} theorems, including inequalities for log concave functions, and with an application to the diffusion equation},
  journal = {Journal of Functional Analysis},
  volume = {22},
  number = {4},
  pages = {366--389},
  year = {1976},
  doi = {10.1016/0022-1236(76)90004-5},
  url = {https://doi.org/10.1016/0022-1236(76)90004-5}
}

@article{BobkovLedoux2000,
  author = {Bobkov, Sergey G. and Ledoux, Michel},
  title = {From {Brunn--Minkowski} to {Brascamp--Lieb} and to logarithmic {Sobolev} inequalities},
  journal = {Geometric and Functional Analysis},
  volume = {10},
  number = {5},
  pages = {1028--1052},
  year = {2000},
  doi = {10.1007/PL00001645},
  url = {https://www.esi.ac.at/preprints/esi819.pdf}
}

@article{PerryWeinBandeiraMoitra2018,
  author = {Perry, Amelia and Wein, Alexander S. and Bandeira, Afonso S. and Moitra, Ankur},
  title = {Optimality and sub-optimality of {PCA} {I}: Spiked random matrix models},
  journal = {The Annals of Statistics},
  volume = {46},
  number = {5},
  pages = {2416--2451},
  year = {2018},
  doi = {10.1214/17-AOS1625},
  eprint = {1807.00891},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1807.00891v2},
  note = {The cited arXiv version includes the supplement as appendices}
}

@article{Eldan2013StochasticLocalization,
  author = {Eldan, Ronen},
  title = {Thin shell implies spectral gap up to polylog via a stochastic localization scheme},
  journal = {Geometric and Functional Analysis},
  volume = {23},
  number = {2},
  pages = {532--569},
  year = {2013},
  doi = {10.1007/s00039-013-0214-y},
  eprint = {1203.0893},
  archivePrefix = {arXiv},
  primaryClass = {math.MG},
  url = {https://arxiv.org/abs/1203.0893v5},
  note = {Proof locators refer to arXiv:1203.0893v5, 16 December 2012}
}

@article{ChenEldan2025Localization,
  author = {Chen, Yuansi and Eldan, Ronen},
  title = {Localization schemes: A framework for proving mixing bounds for {Markov} chains},
  journal = {Duke Mathematical Journal},
  volume = {174},
  number = {8},
  pages = {1431--1510},
  year = {2025},
  doi = {10.1215/00127094-2024-0063},
  eprint = {2203.04163},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  url = {https://arxiv.org/abs/2203.04163v2},
  note = {Proof locators refer to arXiv:2203.04163v2, 6 June 2022}
}

@article{ChewiPooladian2023Entropic,
  author = {Chewi, Sinho and Pooladian, Aram-Alexandre},
  title = {An entropic generalization of {Caffarelli}'s contraction theorem via covariance inequalities},
  journal = {Comptes Rendus. Math{\'e}matique},
  volume = {361},
  number = {G9},
  pages = {1471--1482},
  year = {2023},
  doi = {10.5802/crmath.486},
  url = {https://doi.org/10.5802/crmath.486},
}

@article{Lehec2013Entropy,
  author = {Lehec, Joseph},
  title = {Representation formula for the entropy and functional inequalities},
  journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
  volume = {49},
  number = {3},
  pages = {885--899},
  year = {2013},
  doi = {10.1214/11-AIHP464},
  url = {https://www.numdam.org/articles/10.1214/11-AIHP464/}
}

@article{TAP1977,
  author = {Thouless, David J. and Anderson, Philip W. and Palmer, Richard G.},
  title = {Solution of {'Solvable model of a spin glass'}},
  journal = {Philosophical Magazine},
  volume = {35}, number = {3}, pages = {593--601}, year = {1977},
  doi = {10.1080/14786437708235992},
  url = {https://doi.org/10.1080/14786437708235992}
}

@article{FanMeiMontanari2021,
  author = {Fan, Zhou and Mei, Song and Montanari, Andrea},
  title = {{TAP} free energy, spin glasses, and variational inference},
  journal = {The Annals of Probability},
  volume = {49}, number = {1}, pages = {1--45}, year = {2021},
  doi = {10.1214/20-AOP1443},
  url = {https://arxiv.org/abs/1808.07890v2},
  note = {Proof locators refer to arXiv:1808.07890v2, 26 March 2020}
}

@article{CelentanoFanMei2023,
  author = {Celentano, Michael and Fan, Zhou and Mei, Song},
  title = {Local convexity of the {TAP} free energy and {AMP} convergence for {$\mathbb Z_2$}-synchronization},
  journal = {The Annals of Statistics},
  volume = {51}, number = {2}, pages = {519--546}, year = {2023},
  doi = {10.1214/23-AOS2257},
  url = {https://arxiv.org/abs/2106.11428v3},
  note = {Proof locators refer to arXiv:2106.11428v3, 15 April 2023}
}

@article{SherringtonKirkpatrick1975,
  author = {Sherrington, David and Kirkpatrick, Scott},
  title = {Solvable Model of a Spin-Glass},
  journal = {Physical Review Letters},
  volume = {35}, number = {26}, pages = {1792--1796}, year = {1975},
  doi = {10.1103/PhysRevLett.35.1792},
  url = {https://doi.org/10.1103/PhysRevLett.35.1792}
}

@article{KirkpatrickSherrington1978,
  author = {Kirkpatrick, Scott and Sherrington, David},
  title = {Infinite-ranged models of spin-glasses},
  journal = {Physical Review B},
  volume = {17}, number = {11}, pages = {4384--4403}, year = {1978},
  doi = {10.1103/PhysRevB.17.4384},
  url = {https://doi.org/10.1103/PhysRevB.17.4384}
}

@article{SompolinskyZippelius1982,
  author = {Sompolinsky, Haim and Zippelius, Annette},
  title = {Relaxational dynamics of the {Edwards--Anderson} model and the mean-field theory of spin-glasses},
  journal = {Physical Review B},
  volume = {25}, number = {11}, pages = {6860--6875}, year = {1982},
  doi = {10.1103/PhysRevB.25.6860},
  url = {https://doi.org/10.1103/PhysRevB.25.6860}
}

@article{Billoire2011SlowSamples,
  author = {Billoire, Alain},
  title = {What makes slow samples slow in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Physics A: Mathematical and Theoretical},
  volume = {44}, pages = {075001}, year = {2011},
  eprint = {1011.3420}, archivePrefix = {arXiv},
  note = {arXiv version 2; equation (1), pages 2--3},
  url = {https://arxiv.org/abs/1011.3420v2}
}

@article{BauerschmidtBodineau2019,
  author = {Bauerschmidt, Roland and Bodineau, Thierry},
  title = {A very simple proof of the {LSI} for high temperature spin systems},
  journal = {Journal of Functional Analysis},
  volume = {276}, number = {8}, pages = {2582--2588}, year = {2019},
  doi = {10.1016/j.jfa.2019.01.007},
  eprint = {1712.03676}, archivePrefix = {arXiv}, primaryClass = {math.PR},
  note = {Proof locators refer to arXiv:1712.03676v2, 31 January 2018},
  url = {https://arxiv.org/abs/1712.03676v2}
}

@misc{AnariJainKoehlerPhamVuong2021,
  author = {Anari, Nima and Jain, Vishesh and Koehler, Frederic and Pham, Huy Tuan and Vuong, Thuy-Duong},
  title = {Entropic Independence {I}: Modified Log-{Sobolev} Inequalities for Fractionally Log-Concave Distributions and High-Temperature {Ising} Models},
  year = {2021},
  eprint = {2106.04105}, archivePrefix = {arXiv}, primaryClass = {cs.DS},
  note = {Version 2, 5 November 2021; Theorem 12 and Section 5},
  url = {https://arxiv.org/abs/2106.04105v2}
}

@misc{AcceptedCritical,
  author = {{OpenAI}},
  title = {{Critical slowing down in the Sherrington--Kirkpatrick model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf}{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026}},
  year = {2026},
  note = {Theorems 1.2 and 1.3}
}
@misc{AcceptedGap,
  author = {{OpenAI}},
  title = {{A spectral gap throughout the high-temperature Sherrington--Kirkpatrick phase}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf}{OAI:A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026}},
  year = {2026},
  note = {Lemma 5.7 and Propositions 7.2--7.3; see the hypotheses of Sections 3 and 8}
}

@book{Meckes2019ClassicalGroups,
  author = {Meckes, Elizabeth S.},
  title = {The Random Matrix Theory of the Classical Compact Groups},
  series = {Cambridge Tracts in Mathematics},
  volume = {218},
  publisher = {Cambridge University Press},
  year = {2019},
  doi = {10.1017/9781108303453},
  isbn = {9781108419529},
  url = {https://case.edu/artsci/math/esmeckes/Haar_book.pdf},
  note = {Proof locators refer to the author's final pre-copy-editing draft}
}