\begin{thebibliography}{156}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
  \providecommand{\doi}[1]{doi: #1}\else
  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi

\bibitem[Aldous(1983)]{Aldous1983}
David Aldous.
\newblock Random walks on finite groups and rapidly mixing {Markov} chains.
\newblock In \emph{Séminaire de Probabilités XVII}, volume 986 of
  \emph{Lecture Notes in Mathematics}, pages 243--297. Springer, Berlin, 1983.
\newblock \doi{10.1007/BFb0068322}.

\bibitem[Aldous and Diaconis(1986)]{AldousDiaconis1986}
David Aldous and Persi Diaconis.
\newblock Shuffling cards and stopping times.
\newblock \emph{The American Mathematical Monthly}, 93\penalty0 (5):\penalty0
  333--348, 1986.
\newblock \doi{10.2307/2323590}.

\bibitem[Anari et~al.(2020)Anari, Liu, and
  Oveis~Gharan]{AnariLiuOveisGharan2020}
Nima Anari, Kuikui Liu, and Shayan Oveis~Gharan.
\newblock Spectral independence in high-dimensional expanders and applications
  to the hardcore model.
\newblock In \emph{2020 IEEE 61st Annual Symposium on Foundations of Computer
  Science}, pages 1319--1330. IEEE, 2020.
\newblock \doi{10.1109/FOCS46700.2020.00125}.

\bibitem[Anderson et~al.(2010)Anderson, Guionnet, and
  Zeitouni]{AndersonGuionnetZeitouni2010}
Greg~W. Anderson, Alice Guionnet, and Ofer Zeitouni.
\newblock \emph{An Introduction to Random Matrices}, volume 118 of
  \emph{Cambridge Studies in Advanced Mathematics}.
\newblock Cambridge University Press, 2010.
\newblock \doi{10.1017/CBO9780511801334}.
\newblock URL
  \url{https://assets.cambridge.org/97805211/94525/excerpt/9780521194525_excerpt.pdf}.

\bibitem[Aru and Korzhenkova(2024)]{AruKorzhenkova2024}
Juhan Aru and Aleksandra Korzhenkova.
\newblock The spherical model and the large-n limit of the spin o(n) model via
  the {Gaussian} free field, 2024.
\newblock URL \url{https://arxiv.org/abs/2405.04501v2}.
\newblock Version 2, revised 3 September 2025.

\bibitem[Baik and Lee(2016)]{BaikLee2016}
Jinho Baik and Ji~Oon Lee.
\newblock Fluctuations of the free energy of the spherical
  {Sherrington}-{Kirkpatrick} model.
\newblock \emph{Journal of Statistical Physics}, 165\penalty0 (2):\penalty0
  185--224, 2016.
\newblock \doi{10.1007/s10955-016-1610-0}.
\newblock Locators refer to arXiv:1505.07349v5.

\bibitem[Baik and Lee(2017)]{BaikLee2017}
Jinho Baik and Ji~Oon Lee.
\newblock Fluctuations of the free energy of the spherical
  {Sherrington}-{Kirkpatrick} model with ferromagnetic interaction.
\newblock \emph{Annales Henri Poincare}, 18\penalty0 (6):\penalty0 1867--1917,
  2017.
\newblock \doi{10.1007/s00023-017-0562-5}.

\bibitem[Baik and Lee(2020)]{BaikLee2020}
Jinho Baik and Ji~Oon Lee.
\newblock Free energy of bipartite spherical {Sherrington}-{Kirkpatrick} model.
\newblock \emph{Annales de l'Institut Henri Poincare, Probabilites et
  Statistiques}, 56\penalty0 (4):\penalty0 2897--2934, 2020.
\newblock \doi{10.1214/20-AIHP1062}.

\bibitem[Baik et~al.(2018)Baik, Lee, and Wu]{BaikLeeWu2018}
Jinho Baik, Ji~Oon Lee, and Hao Wu.
\newblock Ferromagnetic to paramagnetic transition in spherical spin glass.
\newblock \emph{Journal of Statistical Physics}, 173\penalty0 (5):\penalty0
  1484--1522, 2018.
\newblock \doi{10.1007/s10955-018-2150-6}.

\bibitem[Baik et~al.(2021)Baik, Collins-Woodfin, Le~Doussal, and
  Wu]{BaikCollinsWoodfinLeDoussalWu2021}
Jinho Baik, Elizabeth Collins-Woodfin, Pierre Le~Doussal, and Hao Wu.
\newblock Spherical spin glass model with external field.
\newblock \emph{Journal of Statistical Physics}, 183\penalty0 (2):\penalty0 31,
  2021.
\newblock \doi{10.1007/s10955-021-02757-7}.

\bibitem[Bakry and {\'E}mery(1985)]{BakryEmery1985}
D.~Bakry and Michel {\'E}mery.
\newblock Diffusions hypercontractives.
\newblock In \emph{S{\'e}minaire de Probabilit{\'e}s XIX, 1983/84}, volume 1123
  of \emph{Lecture Notes in Mathematics}, pages 177--206. Springer, 1985.
\newblock \doi{10.1007/BFb0075847}.

\bibitem[Bakry et~al.(2014)Bakry, Gentil, and Ledoux]{BakryGentilLedoux2014}
Dominique Bakry, Ivan Gentil, and Michel Ledoux.
\newblock \emph{Analysis and Geometry of {Markov} Diffusion Operators}, volume
  348 of \emph{Grundlehren der mathematischen Wissenschaften}.
\newblock Springer, Cham, 2014.
\newblock \doi{10.1007/978-3-319-00227-9}.

\bibitem[Barbier et~al.(2021)Barbier, de~Freitas~Pimenta, Cugliandolo, and
  Stariolo]{BarbierDeFreitasPimentaCugliandoloStariolo2021}
Damien Barbier, Pedro~H. de~Freitas~Pimenta, Leticia~F. Cugliandolo, and
  Daniel~A. Stariolo.
\newblock Finite size effects and loss of self-averageness in the relaxational
  dynamics of the spherical {Sherrington-Kirkpatrick} model.
\newblock \emph{Journal of Statistical Mechanics: Theory and Experiment},
  2021\penalty0 (7):\penalty0 073301, 2021.
\newblock \doi{10.1088/1742-5468/ac0900}.
\newblock Statement locators refer to v2.

\bibitem[Barrera and Jara(2016)]{BarreraJara2016}
Gerardo Barrera and Milton Jara.
\newblock Abrupt convergence for stochastic small perturbations of one
  dimensional dynamical systems.
\newblock \emph{Journal of Statistical Physics}, 163\penalty0 (1):\penalty0
  113--138, 2016.
\newblock \doi{10.1007/s10955-016-1468-1}.

\bibitem[Barrera and Jara(2020)]{BarreraJara2020}
Gerardo Barrera and Milton Jara.
\newblock Thermalisation for small random perturbations of dynamical systems.
\newblock \emph{The Annals of Applied Probability}, 30\penalty0 (3):\penalty0
  1164--1208, 2020.
\newblock \doi{10.1214/19-AAP1526}.

\bibitem[Barrera et~al.(2021{\natexlab{a}})Barrera, Högele, and
  Pardo]{BarreraHogelePardo2021a}
Gerardo Barrera, Michael~A. Högele, and Juan~Carlos Pardo.
\newblock Cutoff thermalization for {Ornstein--Uhlenbeck} systems with small
  {Lévy} noise in the {Wasserstein} distance.
\newblock \emph{Journal of Statistical Physics}, 184:\penalty0 27,
  2021{\natexlab{a}}.
\newblock \doi{10.1007/s10955-021-02815-0}.
\newblock Article 27, 54 pages.

\bibitem[Barrera et~al.(2021{\natexlab{b}})Barrera, Högele, and
  Pardo]{BarreraHogelePardo2021b}
Gerardo Barrera, Michael~A. Högele, and Juan~Carlos Pardo.
\newblock The cutoff phenomenon in total variation for nonlinear {Langevin}
  systems with small layered stable noise.
\newblock \emph{Electronic Journal of Probability}, 26\penalty0 (119):\penalty0
  1--76, 2021{\natexlab{b}}.
\newblock \doi{10.1214/21-EJP685}.

\bibitem[Barthe and Roberto(2003)]{BartheRoberto2003}
F.~Barthe and C.~Roberto.
\newblock {Sobolev} inequalities for probability measures on the real line.
\newblock \emph{Studia Mathematica}, 159\penalty0 (3):\penalty0 481--497, 2003.
\newblock \doi{10.4064/sm159-3-9}.

\bibitem[Bartlett(1951)]{Bartlett1951}
M.~S. Bartlett.
\newblock An inverse matrix adjustment arising in discriminant analysis.
\newblock \emph{The Annals of Mathematical Statistics}, 22\penalty0
  (1):\penalty0 107--111, 1951.
\newblock \doi{10.1214/aoms/1177729698}.

\bibitem[Bauerschmidt and Bodineau(2019)]{BauerschmidtBodineau2019}
Roland Bauerschmidt and Thierry Bodineau.
\newblock A very simple proof of the {LSI} for high temperature spin systems.
\newblock \emph{Journal of Functional Analysis}, 276\penalty0 (8):\penalty0
  2582--2588, 2019.
\newblock \doi{10.1016/j.jfa.2019.01.007}.

\bibitem[Bayer and Diaconis(1992)]{BayerDiaconis1992}
Dave Bayer and Persi Diaconis.
\newblock Trailing the dovetail shuffle to its lair.
\newblock \emph{The Annals of Applied Probability}, 2\penalty0 (2):\penalty0
  294--313, 1992.
\newblock \doi{10.1214/aoap/1177005705}.

\bibitem[Belius and Kistler(2019)]{BeliusKistler2019}
David Belius and Nicola Kistler.
\newblock The {TAP}--{Plefka} variational principle for the spherical {SK}
  model.
\newblock \emph{Communications in Mathematical Physics}, 367:\penalty0
  991--1017, 2019.
\newblock \doi{10.1007/s00220-019-03304-y}.

\bibitem[Ben~Arous et~al.(2006)Ben~Arous, Dembo, and
  Guionnet]{BenArousDemboGuionnet2006}
G{\'e}rard Ben~Arous, Amir Dembo, and Alice Guionnet.
\newblock {Cugliandolo--Kurchan} equations for dynamics of spin-glasses.
\newblock \emph{Probability Theory and Related Fields}, 136\penalty0
  (4):\penalty0 619--660, 2006.
\newblock \doi{10.1007/s00440-005-0491-y}.
\newblock Statement locators refer to v1.

\bibitem[Ben~Arous and Jagannath(2018)]{BenArousJagannath2018}
Gérard Ben~Arous and Aukosh Jagannath.
\newblock Spectral gap estimates in mean field spin glasses.
\newblock \emph{Communications in Mathematical Physics}, 361\penalty0
  (1):\penalty0 1--52, 2018.
\newblock \doi{10.1007/s00220-018-3152-6}.

\bibitem[Ben~Arous and Jagannath(2024)]{BenArousJagannath2024}
Gérard Ben~Arous and Aukosh Jagannath.
\newblock Shattering versus metastability in spin glasses.
\newblock \emph{Communications on Pure and Applied Mathematics}, 77\penalty0
  (1):\penalty0 139--176, 2024.
\newblock \doi{10.1002/cpa.22133}.
\newblock Locators refer to arXiv:2104.08299v1.

\bibitem[Ben~Arous et~al.(2001)Ben~Arous, Dembo, and
  Guionnet]{BenArousDemboGuionnet2001}
Gérard Ben~Arous, Amir Dembo, and Alice Guionnet.
\newblock Aging of spherical spin glasses.
\newblock \emph{Probability Theory and Related Fields}, 120\penalty0
  (1):\penalty0 1--67, 2001.
\newblock \doi{10.1007/PL00008774}.

\bibitem[Ben~Arous et~al.(2020)Ben~Arous, Gheissari, and
  Jagannath]{BenArousGheissariJagannath2020a}
Gérard Ben~Arous, Reza Gheissari, and Aukosh Jagannath.
\newblock Bounding flows for spherical spin glass dynamics.
\newblock \emph{Communications in Mathematical Physics}, 373:\penalty0
  1011--1048, 2020.
\newblock \doi{10.1007/s00220-019-03649-4}.

\bibitem[Berlin and Kac(1952)]{BerlinKac1952}
T.~H. Berlin and M.~Kac.
\newblock The spherical model of a ferromagnet.
\newblock \emph{Physical Review}, 86\penalty0 (6):\penalty0 821--835, 1952.
\newblock \doi{10.1103/PhysRev.86.821}.

\bibitem[Bernstein(1912)]{Bernstein1912}
S.~Bernstein.
\newblock D{\'e}monstration du th{\'e}or{\`e}me de weierstrass fond{\'e}e sur
  le calcul des probabilit{\'e}s.
\newblock \emph{Communications de la Soci{\'e}t{\'e} math{\'e}matique de
  Kharkow, 2e s{\'e}rie}, 13\penalty0 (1):\penalty0 1--2, 1912.
\newblock URL \url{https://www.mathnet.ru/eng/khmo107}.

\bibitem[Billingsley(1999)]{Billingsley1999}
Patrick Billingsley.
\newblock \emph{Convergence of Probability Measures}.
\newblock Wiley Series in Probability and Statistics. John Wiley and Sons, New
  York, 2 edition, 1999.
\newblock \doi{10.1002/9780470316962}.

\bibitem[Bingham(1974)]{Bingham1974}
Christopher Bingham.
\newblock An antipodally symmetric distribution on the sphere.
\newblock \emph{The Annals of Statistics}, 2\penalty0 (6):\penalty0 1201--1225,
  1974.
\newblock \doi{10.1214/aos/1176342874}.

\bibitem[Bismut(1984)]{Bismut1984}
Jean-Michel Bismut.
\newblock \emph{Large Deviations and the Malliavin Calculus}, volume~45 of
  \emph{Progress in Mathematics}.
\newblock Birkh{\"a}user, Boston, 1984.

\bibitem[Blachman(1965)]{Blachman1965}
N.~M. Blachman.
\newblock The convolution inequality for entropy powers.
\newblock \emph{IEEE Transactions on Information Theory}, 11\penalty0
  (2):\penalty0 267--271, 1965.
\newblock \doi{10.1109/TIT.1965.1053768}.

\bibitem[Bogachev(2007)]{Bogachev2007}
Vladimir~I. Bogachev.
\newblock \emph{Measure Theory}, volume 1--2.
\newblock Springer, Berlin, Heidelberg, 2007.
\newblock \doi{10.1007/978-3-540-34514-5}.

\bibitem[Boursier et~al.(2023)Boursier, Chafaï, and
  Labbé]{BoursierChafaiLabbe2023}
Jeanne Boursier, Djalil Chafaï, and Cyril Labbé.
\newblock Universal cutoff for {Dyson Ornstein Uhlenbeck} process.
\newblock \emph{Probability Theory and Related Fields}, 185\penalty0
  (1--2):\penalty0 449--512, 2023.
\newblock \doi{10.1007/s00440-022-01158-5}.
\newblock Locators refer to arXiv:2107.14452v3.

\bibitem[Brascamp and Lieb(1976)]{BrascampLieb1976}
Herm~Jan Brascamp and Elliott~H. Lieb.
\newblock On extensions of the {Brunn--Minkowski} and {Pr\'ekopa--Leindler}
  theorems, including inequalities for log concave functions, and with an
  application to the diffusion equation.
\newblock \emph{Journal of Functional Analysis}, 22\penalty0 (4):\penalty0
  366--389, 1976.
\newblock \doi{10.1016/0022-1236(76)90004-5}.

\bibitem[Brezis(2011)]{Brezis2011}
Haim Brezis.
\newblock \emph{Functional Analysis, {Sobolev} Spaces and Partial Differential
  Equations}.
\newblock Universitext. Springer, New York, 2011.
\newblock \doi{10.1007/978-0-387-70914-7}.

\bibitem[Caltagirone et~al.(2013)Caltagirone, Parisi, and
  Rizzo]{CaltagironeParisiRizzo2013}
Francesco Caltagirone, Giorgio Parisi, and Tommaso Rizzo.
\newblock Critical off-equilibrium dynamics in glassy systems.
\newblock \emph{Physical Review E}, 87\penalty0 (3):\penalty0 032134, 2013.
\newblock \doi{10.1103/PhysRevE.87.032134}.

\bibitem[Cao et~al.(2019)Cao, Lu, and Lu]{CaoLuLu2019}
Yu~Cao, Jianfeng Lu, and Yulong Lu.
\newblock Exponential decay of {R}{\'e}nyi divergence under {F}okker--{P}lanck
  equations.
\newblock \emph{Journal of Statistical Physics}, 176\penalty0 (5):\penalty0
  1172--1184, 2019.
\newblock \doi{10.1007/s10955-019-02339-8}.
\newblock Statement locators refer to v2.

\bibitem[Chafaï and Fathi(2025)]{ChafaiFathi2025}
Djalil Chafaï and Max Fathi.
\newblock On cutoff via rigidity for high dimensional curved diffusions.
\newblock \emph{Comptes Rendus. Mathématique}, 363:\penalty0 1103--1121, 2025.
\newblock \doi{10.5802/crmath.776}.
\newblock Locators refer to the postpublication arXiv version 2412.15969v4.

\bibitem[Chen and Saloff-Coste(2008)]{ChenSaloffCoste2008}
Guan-Yu Chen and Laurent Saloff-Coste.
\newblock The cutoff phenomenon for ergodic {Markov} processes.
\newblock \emph{Electronic Journal of Probability}, 13\penalty0 (3):\penalty0
  26--78, 2008.
\newblock \doi{10.1214/EJP.v13-474}.

\bibitem[Chen et~al.(2011)Chen, Goldstein, and Shao]{ChenGoldsteinShao2011}
Louis H.~Y. Chen, Larry Goldstein, and Qi-Man Shao.
\newblock \emph{Normal Approximation by {S}tein's Method}.
\newblock Probability and Its Applications. Springer, Berlin, Heidelberg, 2011.
\newblock \doi{10.1007/978-3-642-15007-4}.

\bibitem[Chen(2013)]{Chen2013}
Wei-Kuo Chen.
\newblock The aizenman-sims-starr scheme and {Parisi} formula for mixed p-spin
  spherical models.
\newblock \emph{Electronic Journal of Probability}, 18\penalty0 (94):\penalty0
  1--14, 2013.
\newblock \doi{10.1214/EJP.v18-2580}.

\bibitem[Ciuchi and de~Pasquale(1988)]{CiuchiDePasquale1988}
S.~Ciuchi and F.~de~Pasquale.
\newblock Nonlinear relaxation and ergodicity breakdown in random anisotropy
  spin glasses.
\newblock \emph{Nuclear Physics B}, 300:\penalty0 31--60, 1988.
\newblock \doi{10.1016/0550-3213(88)90585-8}.

\bibitem[Collins-Woodfin(2021)]{CollinsWoodfin2021}
Elizabeth Collins-Woodfin.
\newblock Overlaps of a spherical spin glass model with microscopic external
  field.
\newblock \emph{Electronic Journal of Probability}, 26\penalty0 (146):\penalty0
  1--22, 2021.
\newblock \doi{10.1214/21-EJP722}.

\bibitem[Cover and Thomas(2006)]{CoverThomas2006}
Thomas~M. Cover and Joy~A. Thomas.
\newblock \emph{Elements of Information Theory}.
\newblock John Wiley and Sons, 2 edition, 2006.
\newblock \doi{10.1002/047174882X}.

\bibitem[Crisanti and Sommers(1992)]{CrisantiSommers1992}
A.~Crisanti and H.-J. Sommers.
\newblock The spherical p-spin interaction spin glass model: the statics.
\newblock \emph{Zeitschrift fuer Physik B Condensed Matter}, 87\penalty0
  (3):\penalty0 341--354, 1992.
\newblock \doi{10.1007/BF01309287}.

\bibitem[Crisanti et~al.(2019)Crisanti, Sarracino, and
  Zannetti]{CrisantiSarracinoZannetti2019}
A.~Crisanti, A.~Sarracino, and M.~Zannetti.
\newblock Condensation versus ordering: From the spherical models to
  {Bose-Einstein} condensation in the canonical and grand canonical ensemble.
\newblock \emph{Physical Review Research}, 1\penalty0 (2):\penalty0 023022,
  2019.
\newblock \doi{10.1103/PhysRevResearch.1.023022}.

\bibitem[Cuff et~al.(2012)Cuff, Ding, Louidor, Lubetzky, Peres, and
  Sly]{CuffDingLouidorLubetzkyPeresSly2012}
Paul Cuff, Jian Ding, Oren Louidor, Eyal Lubetzky, Yuval Peres, and Allan Sly.
\newblock {Glauber} dynamics for the mean-field {Potts} model.
\newblock \emph{Journal of Statistical Physics}, 149\penalty0 (3):\penalty0
  432--477, 2012.
\newblock \doi{10.1007/s10955-012-0599-2}.

\bibitem[Cugliandolo and Kurchan(1993)]{CugliandoloKurchan1993}
L.~F. Cugliandolo and J.~Kurchan.
\newblock Analytical solution of the off-equilibrium dynamics of a long-range
  spin-glass model.
\newblock \emph{Physical Review Letters}, 71\penalty0 (1):\penalty0 173--176,
  1993.
\newblock \doi{10.1103/PhysRevLett.71.173}.

\bibitem[Cugliandolo and Dean(1995)]{CugliandoloDean1995}
Leticia~F. Cugliandolo and David~S. Dean.
\newblock Full dynamical solution for a spherical spin-glass model.
\newblock \emph{Journal of Physics A: Mathematical and General}, 28\penalty0
  (15):\penalty0 4213--4234, 1995.
\newblock \doi{10.1088/0305-4470/28/15/003}.
\newblock Locators and preprint page numbers refer to arXiv:cond-mat/9502075v1.

\bibitem[Cugliandolo et~al.(1997)Cugliandolo, Kurchan, and
  Peliti]{CugliandoloKurchanPeliti1997}
Leticia~F. Cugliandolo, Jorge Kurchan, and Luca Peliti.
\newblock Energy flow, partial equilibration, and effective temperatures in
  systems with slow dynamics.
\newblock \emph{Physical Review E}, 55\penalty0 (4):\penalty0 3898--3914, 1997.
\newblock \doi{10.1103/PhysRevE.55.3898}.

\bibitem[Cugliandolo et~al.(2007)Cugliandolo, Dean, and
  Yoshino]{CugliandoloDeanYoshino2007}
Leticia~F. Cugliandolo, David~S. Dean, and Hajime Yoshino.
\newblock Nonlinear susceptibilities of spherical models.
\newblock \emph{Journal of Physics A: Mathematical and Theoretical},
  40\penalty0 (16):\penalty0 4285--4303, 2007.
\newblock \doi{10.1088/1751-8113/40/16/003}.

\bibitem[Cugliandolo et~al.(2018)Cugliandolo, Lozano, Nessi, Picco, and
  Tartaglia]{CugliandoloLozanoNessiPiccoTartaglia2018}
Leticia~F. Cugliandolo, Gustavo~S. Lozano, Nicol{\'a}s Nessi, Marco Picco, and
  Alessandro Tartaglia.
\newblock Quenched dynamics of classical isolated systems: the spherical spin
  model with two-body random interactions or the {Neumann} integrable model.
\newblock \emph{Journal of Statistical Mechanics: Theory and Experiment},
  2018\penalty0 (6):\penalty0 063206, 2018.
\newblock \doi{10.1088/1742-5468/aac2fe}.

\bibitem[Dembo et~al.(2007)Dembo, Guionnet, and Mazza]{DemboGuionnetMazza2007}
Amir Dembo, Alice Guionnet, and Christian Mazza.
\newblock Limiting dynamics for spherical models of spin glasses at high
  temperature.
\newblock \emph{Journal of Statistical Physics}, 128\penalty0 (4):\penalty0
  847--881, 2007.
\newblock \doi{10.1007/s10955-006-9239-z}.
\newblock Locators refer to arXiv:math/0609546v1.

\bibitem[Diaconis(1996)]{Diaconis1996}
Persi Diaconis.
\newblock The cutoff phenomenon in finite {Markov} chains.
\newblock \emph{Proceedings of the National Academy of Sciences of the USA},
  93\penalty0 (4):\penalty0 1659--1664, 1996.
\newblock \doi{10.1073/pnas.93.4.1659}.

\bibitem[Diaconis and Freedman(1987)]{DiaconisFreedman1987}
Persi Diaconis and David Freedman.
\newblock A dozen de finetti-style results in search of a theory.
\newblock \emph{Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et
  Statistiques}, 23\penalty0 (S2):\penalty0 397--423, 1987.
\newblock URL \url{https://www.numdam.org/item/AIHPB_1987__23_S2_397_0/}.

\bibitem[Diaconis and Shahshahani(1981)]{DiaconisShahshahani1981}
Persi Diaconis and Mehrdad Shahshahani.
\newblock Generating a random permutation with random transpositions.
\newblock \emph{Zeitschrift für Wahrscheinlichkeitstheorie und verwandte
  Gebiete}, 57\penalty0 (2):\penalty0 159--179, 1981.
\newblock \doi{10.1007/BF00535487}.

\bibitem[Diaconis et~al.(1990)Diaconis, Graham, and
  Morrison]{DiaconisGrahamMorrison1990}
Persi Diaconis, R.~L. Graham, and J.~A. Morrison.
\newblock Asymptotic analysis of a random walk on a hypercube with many
  dimensions.
\newblock \emph{Random Structures \& Algorithms}, 1\penalty0 (1):\penalty0
  51--72, 1990.
\newblock \doi{10.1002/rsa.3240010105}.

\bibitem[Ding et~al.(2009{\natexlab{a}})Ding, Lubetzky, and
  Peres]{DingLubetzkyPeres2009a}
Jian Ding, Eyal Lubetzky, and Yuval Peres.
\newblock The mixing time evolution of {Glauber} dynamics for the mean-field
  {Ising} model.
\newblock \emph{Communications in Mathematical Physics}, 289\penalty0
  (2):\penalty0 725--764, 2009{\natexlab{a}}.
\newblock \doi{10.1007/s00220-009-0781-9}.

\bibitem[Ding et~al.(2009{\natexlab{b}})Ding, Lubetzky, and
  Peres]{DingLubetzkyPeres2009b}
Jian Ding, Eyal Lubetzky, and Yuval Peres.
\newblock Censored {Glauber} dynamics for the mean field {Ising} model.
\newblock \emph{Journal of Statistical Physics}, 137\penalty0 (3):\penalty0
  407--458, 2009{\natexlab{b}}.
\newblock \doi{10.1007/s10955-009-9859-1}.

\bibitem[Du and Huang(2026)]{DuHuang2026}
Hang Du and Brice Huang.
\newblock Overlap distribution of the critical {Sherrington}-{Kirkpatrick}
  model, 2026.
\newblock URL \url{https://arxiv.org/abs/2608.08752v1}.
\newblock Version 1; preprint.

\bibitem[Dumitriu and Edelman(2002)]{DumitriuEdelman2002}
Ioana Dumitriu and Alan Edelman.
\newblock Matrix models for beta ensembles.
\newblock \emph{Journal of Mathematical Physics}, 43\penalty0 (11):\penalty0
  5830--5847, 2002.
\newblock \doi{10.1063/1.1507823}.

\bibitem[Edelman and Sutton(2007)]{EdelmanSutton2007}
Alan Edelman and Brian~D. Sutton.
\newblock From random matrices to stochastic operators.
\newblock \emph{Journal of Statistical Physics}, 127\penalty0 (6):\penalty0
  1121--1165, 2007.
\newblock \doi{10.1007/s10955-006-9226-4}.

\bibitem[El~Alaoui et~al.(2022)El~Alaoui, Montanari, and
  Sellke]{ElAlaouiMontanariSellke2022}
Ahmed El~Alaoui, Andrea Montanari, and Mark Sellke.
\newblock Sampling from the {Sherrington--Kirkpatrick} {Gibbs} measure via
  algorithmic stochastic localization.
\newblock In \emph{2022 IEEE 63rd Annual Symposium on Foundations of Computer
  Science}, pages 323--334. IEEE, 2022.
\newblock \doi{10.1109/FOCS54457.2022.00038}.

\bibitem[Eldan et~al.(2022)Eldan, Koehler, and
  Zeitouni]{EldanKoehlerZeitouni2022}
Ronen Eldan, Frederic Koehler, and Ofer Zeitouni.
\newblock A spectral condition for spectral gap: fast mixing in
  high-temperature {Ising} models.
\newblock \emph{Probability Theory and Related Fields}, 182\penalty0
  (3--4):\penalty0 1035--1051, 2022.
\newblock \doi{10.1007/s00440-021-01085-x}.

\bibitem[Elworthy and Li(1994)]{ElworthyLi1994}
K.~D. Elworthy and Xue-Mei Li.
\newblock Formulae for the derivatives of heat semigroups.
\newblock \emph{Journal of Functional Analysis}, 125\penalty0 (1):\penalty0
  252--286, 1994.
\newblock \doi{10.1006/jfan.1994.1124}.
\newblock Author-posted version of the 1994 article; locators refer to
  arXiv:1911.10971v1.

\bibitem[Engel and Nagel(2000)]{EngelNagel2000}
Klaus-Jochen Engel and Rainer Nagel.
\newblock \emph{One-Parameter Semigroups for Linear Evolution Equations},
  volume 194 of \emph{Graduate Texts in Mathematics}.
\newblock Springer, New York, 2000.
\newblock \doi{10.1007/b97696}.

\bibitem[Erd{\H o}s et~al.(2012)Erd{\H o}s, Yau, and Yin]{ErdosYauYin2012}
L{\'a}szl{\'o} Erd{\H o}s, Horng-Tzer Yau, and Jun Yin.
\newblock Rigidity of eigenvalues of generalized {Wigner} matrices.
\newblock \emph{Advances in Mathematics}, 229\penalty0 (3):\penalty0
  1435--1515, 2012.
\newblock \doi{10.1016/j.aim.2011.12.010}.
\newblock Locators refer to arXiv version 7.

\bibitem[Farina and Rinaldi(2000)]{FarinaRinaldi2000}
Lorenzo Farina and Sergio Rinaldi.
\newblock \emph{Positive Linear Systems: Theory and Applications}.
\newblock John Wiley and Sons, 2000.
\newblock \doi{10.1002/9781118033029}.

\bibitem[Feller(1971)]{Feller1971}
William Feller.
\newblock \emph{An Introduction to Probability Theory and Its Applications},
  volume~2.
\newblock John Wiley and Sons, New York, 2 edition, 1971.

\bibitem[Forrester and Rains(2001)]{ForresterRains2001}
Peter~J. Forrester and Eric~M. Rains.
\newblock Interrelationships between orthogonal, unitary and symplectic matrix
  ensembles.
\newblock In Pavel Bleher and Alexander Its, editors, \emph{Random Matrix
  Models and Their Applications}, volume~40 of \emph{Mathematical Sciences
  Research Institute Publications}, pages 171--208. Cambridge University Press,
  2001.

\bibitem[Fyodorov et~al.(2015)Fyodorov, Perret, and
  Schehr]{FyodorovPerretSchehr2015}
Yan~V. Fyodorov, Anthony Perret, and Gr{\'e}gory Schehr.
\newblock Large time zero temperature dynamics of the spherical {$p=2$}-spin
  glass model of finite size.
\newblock \emph{Journal of Statistical Mechanics: Theory and Experiment},
  2015\penalty0 (11):\penalty0 P11017, 2015.
\newblock \doi{10.1088/1742-5468/2015/11/P11017}.

\bibitem[Ge et~al.(2021)Ge, Lee, Lu, and Risteski]{GeLeeLuRisteski2021}
Rong Ge, Holden Lee, Jianfeng Lu, and Andrej Risteski.
\newblock Efficient sampling from the {Bingham} distribution.
\newblock In Vitaly Feldman, Katrina Ligett, and Sivan Sabato, editors,
  \emph{Proceedings of the 32nd International Conference on Algorithmic
  Learning Theory}, volume 132 of \emph{Proceedings of Machine Learning
  Research}, pages 673--685. PMLR, 2021.
\newblock URL \url{https://proceedings.mlr.press/v132/ge21a.html}.

\bibitem[Genovese and Tantari(2015)]{GenoveseTantari2015}
Giuseppe Genovese and Daniele Tantari.
\newblock Legendre duality of spherical and {Gaussian} spin glasses.
\newblock \emph{Mathematical Physics, Analysis and Geometry}, 18:\penalty0 10,
  2015.
\newblock \doi{10.1007/s11040-015-9181-x}.

\bibitem[Gentil(2010)]{Gentil2010}
Ivan Gentil.
\newblock Logarithmic {Sobolev} inequality for diffusion semigroups, 2010.
\newblock Lecture notes; locators refer to v1.

\bibitem[Gheissari and Jagannath(2019)]{GheissariJagannath2019}
Reza Gheissari and Aukosh Jagannath.
\newblock On the spectral gap of spherical spin glass dynamics.
\newblock \emph{Annales de l'Institut Henri Poincaré, Probabilités et
  Statistiques}, 55\penalty0 (2):\penalty0 756--776, 2019.
\newblock \doi{10.1214/18-AIHP897}.

\bibitem[Gheissari et~al.(2026)Gheissari, Grant, and Yu]{GheissariGrantYu2026}
Reza Gheissari, Curtis Grant, and Tianmin Yu.
\newblock Mixing times of {Langevin} dynamics for spiked matrix models, 2026.
\newblock URL \url{https://arxiv.org/abs/2604.20008v2}.
\newblock Version 2, 20 May 2026.

\bibitem[Gross(1975)]{Gross1975}
Leonard Gross.
\newblock Logarithmic {Sobolev} inequalities.
\newblock \emph{American Journal of Mathematics}, 97\penalty0 (4):\penalty0
  1061--1083, 1975.
\newblock \doi{10.2307/2373688}.

\bibitem[Gustavsson(2005)]{Gustavsson2005}
Jonas Gustavsson.
\newblock {Gaussian} fluctuations of eigenvalues in the {GUE}.
\newblock \emph{Annales de l'Institut Henri Poincare, Probabilites et
  Statistiques}, 41\penalty0 (2):\penalty0 151--178, 2005.
\newblock \doi{10.1016/j.anihpb.2004.04.002}.

\bibitem[Hausdorff(1921{\natexlab{a}})]{Hausdorff1921a}
Felix Hausdorff.
\newblock Summationsmethoden und momentfolgen. i.
\newblock \emph{Mathematische Zeitschrift}, 9\penalty0 (1--2):\penalty0
  74--109, 1921{\natexlab{a}}.
\newblock \doi{10.1007/BF01378337}.

\bibitem[Hausdorff(1921{\natexlab{b}})]{Hausdorff1921b}
Felix Hausdorff.
\newblock Summationsmethoden und momentfolgen. ii.
\newblock \emph{Mathematische Zeitschrift}, 9\penalty0 (3--4):\penalty0
  280--299, 1921{\natexlab{b}}.
\newblock \doi{10.1007/BF01279032}.

\bibitem[Hermon(2015)]{Hermon2015}
Jonathan Hermon.
\newblock A technical report on hitting times, mixing and cutoff, 2015.
\newblock URL \url{https://arxiv.org/abs/1501.01869v6}.
\newblock Version 6, 18 January 2018.

\bibitem[Holley and Stroock(1987)]{HolleyStroock1987}
Richard Holley and Daniel Stroock.
\newblock Logarithmic {Sobolev} inequalities and stochastic {Ising} models.
\newblock \emph{Journal of Statistical Physics}, 46\penalty0 (5-6):\penalty0
  1159--1194, 1987.
\newblock \doi{10.1007/BF01011161}.

\bibitem[H{\"o}rmander(1967)]{Hormander1967}
Lars H{\"o}rmander.
\newblock Hypoelliptic second order differential equations.
\newblock \emph{Acta Mathematica}, 119:\penalty0 147--171, 1967.
\newblock \doi{10.1007/BF02392081}.

\bibitem[Hsu et~al.(2012)Hsu, Kakade, and Zhang]{HsuKakadeZhang2012}
Daniel Hsu, Sham~M. Kakade, and Tong Zhang.
\newblock A tail inequality for quadratic forms of subgaussian random vectors.
\newblock \emph{Electronic Communications in Probability}, 17\penalty0
  (52):\penalty0 1--6, 2012.
\newblock \doi{10.1214/ECP.v17-2079}.
\newblock URL \url{https://arxiv.org/abs/1110.2842v1}.
\newblock Statement locators refer to arXiv version 1.

\bibitem[Hsu(2002)]{Hsu2002}
Elton~P. Hsu.
\newblock \emph{Stochastic Analysis on Manifolds}, volume~38 of \emph{Graduate
  Studies in Mathematics}.
\newblock American Mathematical Society, Providence, RI, 2002.
\newblock \doi{10.1090/gsm/038}.

\bibitem[Huang et~al.(2024)Huang, Montanari, and Pham]{HuangMontanariPham2024}
Brice Huang, Andrea Montanari, and Huy~Tuan Pham.
\newblock Sampling from spherical spin glasses in total variation via
  algorithmic stochastic localization, 2024.
\newblock URL \url{https://arxiv.org/abs/2404.15651v1}.
\newblock Version 1, 24 April 2024.

\bibitem[Huang et~al.(2025)Huang, Mohanty, Rajaraman, and
  Wu]{HuangMohantyRajaramanWu2025}
Brice Huang, Sidhanth Mohanty, Amit Rajaraman, and David~X. Wu.
\newblock Weak {Poincaré} inequalities, simulated annealing, and sampling from
  spherical spin glasses.
\newblock In \emph{Proceedings of the 57th Annual ACM Symposium on Theory of
  Computing}, pages 915--923. ACM, 2025.
\newblock \doi{10.1145/3717823.3718236}.
\newblock Locators refer to arXiv:2411.09075v2.

\bibitem[Jain and Mizgerd(2026)]{JainMizgerd2026}
Vishesh Jain and Clayton Mizgerd.
\newblock Total variation cutoff for {Kac}'s walk on the sphere, 2026.
\newblock URL \url{https://arxiv.org/abs/2607.13401v1}.
\newblock Version 1, 15 July 2026.

\bibitem[Johnstone et~al.(2024)Johnstone, Klochkov, Onatski, and
  Pavlyshyn]{JohnstoneKlochkovOnatskiPavlyshyn2024}
Iain~M. Johnstone, Yegor Klochkov, Alexei Onatski, and Damian Pavlyshyn.
\newblock Spin glass to paramagnetic transition and triple point in spherical
  {SK} model.
\newblock \emph{Journal of Statistical Physics}, 191:\penalty0 99, 2024.
\newblock \doi{10.1007/s10955-024-03296-7}.
\newblock Locators refer to arXiv:2104.07629v4.

\bibitem[Kac and Thompson(1977)]{KacThompson1977}
Mark Kac and Colin~J. Thompson.
\newblock Correlation functions in the spherical and mean spherical models.
\newblock \emph{Journal of Mathematical Physics}, 18\penalty0 (8):\penalty0
  1650--1653, 1977.
\newblock \doi{10.1063/1.523452}.

\bibitem[Kantorovich(1948)]{Kantorovich1948}
L.~V. Kantorovich.
\newblock Functional analysis and applied mathematics.
\newblock \emph{Uspekhi Matematicheskikh Nauk}, 3\penalty0 (6(28)):\penalty0
  89--185, 1948.
\newblock URL
  \url{https://www.mathnet.ru/php/archive.phtml?jrnid=rm&option_lang=eng&paperid=8775&wshow=paper}.

\bibitem[Kato(1995)]{Kato1995}
Tosio Kato.
\newblock \emph{Perturbation Theory for Linear Operators}.
\newblock Classics in Mathematics. Springer, Berlin, Heidelberg, 2 edition,
  1995.
\newblock \doi{10.1007/978-3-642-66282-9}.
\newblock Reprint of the 1980 edition.

\bibitem[Khas'minskii(1959)]{Khasminskii1959}
R.~Z. Khas'minskii.
\newblock On positive solutions of the equation {$\mathfrak{A}u+Vu=0$}.
\newblock \emph{Teoriya Veroyatnostei i ee Primeneniya}, 4\penalty0
  (3):\penalty0 332--341, 1959.
\newblock \doi{10.1137/1104030}.
\newblock English translation: Theory of Probability and its Applications 4
  (1959), no. 3, pp.~309--318.

\bibitem[Kolesnikov and Milman(2017)]{KolesnikovMilman2017}
Alexander~V. Kolesnikov and Emanuel Milman.
\newblock {Brascamp--Lieb}-type inequalities on weighted {Riemannian} manifolds
  with boundary.
\newblock \emph{The Journal of Geometric Analysis}, 27\penalty0 (2):\penalty0
  1680--1702, 2017.
\newblock \doi{10.1007/s12220-016-9736-5}.
\newblock Locators refer to arXiv:1310.2526v7.

\bibitem[Kosterlitz et~al.(1976)Kosterlitz, Thouless, and
  Jones]{KosterlitzThoulessJones1976}
J.~M. Kosterlitz, D.~J. Thouless, and Raymund~C. Jones.
\newblock Spherical model of a spin-glass.
\newblock \emph{Physical Review Letters}, 36\penalty0 (20):\penalty0
  1217--1220, 1976.
\newblock \doi{10.1103/PhysRevLett.36.1217}.

\bibitem[Kume and Wood(2005)]{KumeWood2005}
A.~Kume and Andrew T.~A. Wood.
\newblock Saddlepoint approximations for the {Bingham} and {Fisher}--{Bingham}
  normalising constants.
\newblock \emph{Biometrika}, 92\penalty0 (2):\penalty0 465--476, 2005.
\newblock \doi{10.1093/biomet/92.2.465}.

\bibitem[Lachaud(2005)]{Lachaud2005}
Béatrice Lachaud.
\newblock Cut-off and hitting times of a sample of {Ornstein--Uhlenbeck}
  processes and its average.
\newblock \emph{Journal of Applied Probability}, 42\penalty0 (4):\penalty0
  1069--1080, 2005.
\newblock \doi{10.1239/jap/1134587817}.

\bibitem[Lacoin(2016)]{Lacoin2016}
Hubert Lacoin.
\newblock The cutoff profile for the simple exclusion process on the circle.
\newblock \emph{The Annals of Probability}, 44\penalty0 (5):\penalty0
  3399--3430, 2016.
\newblock \doi{10.1214/15-AOP1053}.

\bibitem[Landon(2022)]{Landon2022}
Benjamin Landon.
\newblock Free energy fluctuations of the two-spin spherical {SK} model at
  critical temperature.
\newblock \emph{Journal of Mathematical Physics}, 63\penalty0 (3):\penalty0
  033301, 2022.
\newblock \doi{10.1063/5.0054298}.
\newblock Locators refer to arXiv version 1.

\bibitem[Landon and Sosoe(2022)]{LandonSosoe2022}
Benjamin Landon and Philippe Sosoe.
\newblock Fluctuations of the overlap at low temperature in the 2-spin
  spherical {SK} model.
\newblock \emph{Annales de l'Institut Henri Poincare, Probabilites et
  Statistiques}, 58\penalty0 (3):\penalty0 1426--1459, 2022.
\newblock \doi{10.1214/21-AIHP1205}.
\newblock Locators refer to arXiv version 1.

\bibitem[Landon and Sosoe(2025)]{LandonSosoe2025}
Benjamin Landon and Philippe Sosoe.
\newblock Fluctuations of the 2-spin {SSK} model with magnetic field.
\newblock \emph{The Annals of Applied Probability}, 35\penalty0 (5):\penalty0
  3543--3619, 2025.
\newblock \doi{10.1214/25-AAP2203}.

\bibitem[Le~Cam(1960)]{LeCam1960}
Lucien Le~Cam.
\newblock Locally asymptotically normal families of distributions.
\newblock \emph{University of California Publications in Statistics},
  3\penalty0 (2):\penalty0 37--98, 1960.

\bibitem[Levin and Peres(2017)]{LevinPeres2017}
David~A. Levin and Yuval Peres.
\newblock \emph{{Markov} Chains and Mixing Times}.
\newblock American Mathematical Society, Providence, RI, 2 edition, 2017.
\newblock ISBN 9781470429621.
\newblock \doi{10.1090/mbk/107}.
\newblock URL \url{https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf}.
\newblock With contributions by Elizabeth L. Wilmer.

\bibitem[Levin et~al.(2010)Levin, Luczak, and Peres]{LevinLuczakPeres2010}
David~A. Levin, Malwina~J. Luczak, and Yuval Peres.
\newblock {Glauber} dynamics for the mean-field {Ising} model: cut-off,
  critical power law, and metastability.
\newblock \emph{Probability Theory and Related Fields}, 146\penalty0
  (1--2):\penalty0 223--265, 2010.
\newblock \doi{10.1007/s00440-008-0189-z}.

\bibitem[Lewis and Wannier(1952)]{LewisWannier1952}
H.~W. Lewis and G.~H. Wannier.
\newblock Spherical model of a ferromagnet.
\newblock \emph{Physical Review}, 88\penalty0 (3):\penalty0 682--683, 1952.
\newblock \doi{10.1103/PhysRev.88.682.2}.

\bibitem[Lubetzky and Sly(2016)]{LubetzkySly2016}
Eyal Lubetzky and Allan Sly.
\newblock Information percolation and cutoff for the stochastic {Ising} model.
\newblock \emph{Journal of the American Mathematical Society}, 29\penalty0
  (3):\penalty0 729--774, 2016.
\newblock \doi{10.1090/jams/841}.

\bibitem[Lukkarinen(2020)]{Lukkarinen2020}
Jani Lukkarinen.
\newblock Multi-state condensation in {Berlin}--{Kac} spherical models.
\newblock \emph{Communications in Mathematical Physics}, 373\penalty0
  (1):\penalty0 389--433, 2020.
\newblock \doi{10.1007/s00220-019-03659-2}.

\bibitem[Marsaglia(1965)]{Marsaglia1965}
George Marsaglia.
\newblock Ratios of normal variables and ratios of sums of uniform variables.
\newblock \emph{Journal of the American Statistical Association}, 60\penalty0
  (309):\penalty0 193--204, 1965.
\newblock \doi{10.1080/01621459.1965.10480783}.

\bibitem[Metzler(1945)]{Metzler1945}
Lloyd~A. Metzler.
\newblock Stability of multiple markets: The hicks conditions.
\newblock \emph{Econometrica}, 13\penalty0 (4):\penalty0 277--292, 1945.
\newblock \doi{10.2307/1906922}.

\bibitem[Moslehian(2013)]{Moslehian2013}
Mohammad~Sal Moslehian.
\newblock Operator {Kantorovich} type inequality.
\newblock \emph{RIMS K{\^o}ky{\^u}roku}, 1839:\penalty0 119--131, 2013.
\newblock URL
  \url{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1839-16.pdf}.

\bibitem[Méliot(2014)]{Meliot2014}
Pierre-Loïc Méliot.
\newblock The cut-off phenomenon for {Brownian} motions on compact symmetric
  spaces.
\newblock \emph{Potential Analysis}, 40\penalty0 (4):\penalty0 427--509, 2014.
\newblock \doi{10.1007/s11118-013-9356-7}.

\bibitem[Nestoridi and Olesker-Taylor(2022)]{NestoridiOleskerTaylor2022}
Evita Nestoridi and Sam Olesker-Taylor.
\newblock Limit profiles for reversible {Markov} chains.
\newblock \emph{Probability Theory and Related Fields}, 182\penalty0
  (1--2):\penalty0 157--188, 2022.
\newblock \doi{10.1007/s00440-021-01061-5}.

\bibitem[Nguyen and Sosoe(2019)]{NguyenSosoe2019}
Vu~Lan Nguyen and Philippe Sosoe.
\newblock Central limit theorem near the critical temperature for the overlap
  in the 2-spin spherical {SK} model.
\newblock \emph{Journal of Mathematical Physics}, 60\penalty0 (10):\penalty0
  103302, 2019.
\newblock \doi{10.1063/1.5065525}.
\newblock Locators refer to arXiv:1809.03675v1.

\bibitem[Nourdin and Peccati(2012)]{NourdinPeccati2012}
Ivan Nourdin and Giovanni Peccati.
\newblock \emph{Normal Approximations with Malliavin Calculus: From {Stein}'s
  Method to Universality}, volume 192 of \emph{Cambridge Tracts in
  Mathematics}.
\newblock Cambridge University Press, 2012.
\newblock \doi{10.1017/CBO9781139084659}.

\bibitem[Olver et~al.(2010)Olver, Lozier, Boisvert, and
  Clark]{OlverLozierBoisvertClark2010}
Frank W.~J. Olver, Daniel~W. Lozier, Ronald~F. Boisvert, and Charles~W. Clark,
  editors.
\newblock \emph{NIST Handbook of Mathematical Functions}.
\newblock Cambridge University Press, 2010.
\newblock ISBN 978-0-521-19225-5.

\bibitem[Panchenko and Talagrand(2007)]{PanchenkoTalagrand2007}
Dmitry Panchenko and Michel Talagrand.
\newblock On the overlap in the multiple spherical {SK} models.
\newblock \emph{The Annals of Probability}, 35\penalty0 (6):\penalty0
  2321--2355, 2007.
\newblock \doi{10.1214/009117907000000015}.

\bibitem[Pedrotti and Salez(2026)]{PedrottiSalez2026}
Francesco Pedrotti and Justin Salez.
\newblock The local product condition implies cutoff, 2026.
\newblock URL \url{https://arxiv.org/abs/2607.05345v2}.
\newblock Version 2, 13 July 2026.

\bibitem[Pillai and Smith(2017)]{PillaiSmith2017}
Natesh~S. Pillai and Aaron Smith.
\newblock {Kac}'s walk on $n$-sphere mixes in $n\log n$ steps.
\newblock \emph{The Annals of Applied Probability}, 27\penalty0 (1):\penalty0
  631--650, 2017.
\newblock \doi{10.1214/16-AAP1214}.

\bibitem[Ram{\'i}rez et~al.(2011)Ram{\'i}rez, Rider, and
  Vir{\'a}g]{RamirezRiderVirag2011}
Jos{\'e}~A. Ram{\'i}rez, Brian Rider, and B{\'a}lint Vir{\'a}g.
\newblock Beta ensembles, stochastic {Airy} spectrum, and a diffusion.
\newblock \emph{Journal of the American Mathematical Society}, 24\penalty0
  (4):\penalty0 919--944, 2011.
\newblock \doi{10.1090/S0894-0347-2011-00703-0}.
\newblock Locators refer to arXiv version 4.

\bibitem[R{\'e}nyi(1961)]{Renyi1961}
Alfr{\'e}d R{\'e}nyi.
\newblock On measures of entropy and information.
\newblock In \emph{Proceedings of the Fourth Berkeley Symposium on Mathematical
  Statistics and Probability}, volume~1, pages 547--561, Berkeley, 1961.
  University of California Press.

\bibitem[Revuz and Yor(1999)]{RevuzYor1999}
Daniel Revuz and Marc Yor.
\newblock \emph{Continuous Martingales and {Brownian} Motion}, volume 293 of
  \emph{Grundlehren der mathematischen Wissenschaften}.
\newblock Springer, Berlin, 3 edition, 1999.
\newblock \doi{10.1007/978-3-662-06400-9}.

\bibitem[Rodgers and Moore(1989)]{RodgersMoore1989}
G.~J. Rodgers and M.~A. Moore.
\newblock Distribution of barrier heights in infinite-range spin glass models.
\newblock \emph{Journal of Physics A: Mathematical and General}, 22\penalty0
  (8):\penalty0 1085--1100, 1989.
\newblock \doi{10.1088/0305-4470/22/8/022}.

\bibitem[Rothaus(1985)]{Rothaus1985}
O.~S. Rothaus.
\newblock Analytic inequalities, isoperimetric inequalities and logarithmic
  {Sobolev} inequalities.
\newblock \emph{Journal of Functional Analysis}, 64\penalty0 (2):\penalty0
  296--313, 1985.
\newblock \doi{10.1016/0022-1236(85)90079-5}.

\bibitem[Salez(2024)]{Salez2024}
Justin Salez.
\newblock Cutoff for non-negatively curved {Markov} chains.
\newblock \emph{Journal of the European Mathematical Society}, 26\penalty0
  (11):\penalty0 4375--4392, 2024.
\newblock \doi{10.4171/JEMS/1348}.

\bibitem[Salez(2026)]{Salez2026}
Justin Salez.
\newblock Cutoff for non-negatively curved diffusions.
\newblock \emph{Inventiones mathematicae}, 246\penalty0 (1):\penalty0 1--11,
  2026.
\newblock \doi{10.1007/s00222-026-01431-7}.

\bibitem[Saloff-Coste(1994)]{SaloffCoste1994}
Laurent Saloff-Coste.
\newblock Precise estimates on the rate at which certain diffusions tend to
  equilibrium.
\newblock \emph{Mathematische Zeitschrift}, 217:\penalty0 641--677, 1994.
\newblock \doi{10.1007/BF02571965}.

\bibitem[Scheff{\'e}(1947)]{Scheffe1947}
Henry Scheff{\'e}.
\newblock A useful convergence theorem for probability distributions.
\newblock \emph{The Annals of Mathematical Statistics}, 18\penalty0
  (3):\penalty0 434--438, 1947.
\newblock \doi{10.1214/aoms/1177730390}.

\bibitem[Setti(1998)]{Setti1998}
Alberto~G. Setti.
\newblock Eigenvalue estimates for the weighted {Laplacian} on a {Riemannian}
  manifold.
\newblock \emph{Rendiconti del Seminario Matematico della Universit{\`a} di
  Padova}, 100:\penalty0 27--55, 1998.
\newblock URL \url{https://www.numdam.org/item/RSMUP_1998__100__27_0/}.

\bibitem[Sherman and Morrison(1950)]{ShermanMorrison1950}
Jack Sherman and Winifred~J. Morrison.
\newblock Adjustment of an inverse matrix corresponding to a change in one
  element of a given matrix.
\newblock \emph{The Annals of Mathematical Statistics}, 21\penalty0
  (1):\penalty0 124--127, 1950.
\newblock \doi{10.1214/aoms/1177729893}.

\bibitem[Sherrington and Kirkpatrick(1975)]{SherringtonKirkpatrick1975}
David Sherrington and Scott Kirkpatrick.
\newblock Solvable model of a spin-glass.
\newblock \emph{Physical Review Letters}, 35\penalty0 (26):\penalty0
  1792--1796, 1975.
\newblock \doi{10.1103/PhysRevLett.35.1792}.

\bibitem[Shukla and Singh(1981)]{ShuklaSingh1981}
Prabodh Shukla and Surjit Singh.
\newblock Spin glass dynamics in the spherical model.
\newblock \emph{Journal of Physics C: Solid State Physics}, 14\penalty0
  (4):\penalty0 L81--L84, 1981.
\newblock \doi{10.1088/0022-3719/14/4/004}.

\bibitem[Simon(2014)]{Simon2014}
Leon Simon.
\newblock Introduction to geometric measure theory, 2014.
\newblock URL \url{https://web.stanford.edu/class/math285/ts-gmt.pdf}.
\newblock Author lecture notes.

\bibitem[Sokal(2010)]{Sokal2010}
Alan~D. Sokal.
\newblock Real-variables characterization of generalized {Stieltjes} functions.
\newblock \emph{Expositiones Mathematicae}, 28\penalty0 (2):\penalty0 179--185,
  2010.
\newblock \doi{10.1016/j.exmath.2009.06.004}.
\newblock URL \url{https://arxiv.org/abs/0902.0065v2}.
\newblock Equation locators refer to arXiv version 2.

\bibitem[Soshnikov(1999)]{Soshnikov1999}
Alexander Soshnikov.
\newblock Universality at the edge of the spectrum in {Wigner} random matrices.
\newblock \emph{Communications in Mathematical Physics}, 207\penalty0
  (3):\penalty0 697--733, 1999.
\newblock \doi{10.1007/s002200050743}.

\bibitem[Soshnikov(2000)]{Soshnikov2000}
Alexander~B. Soshnikov.
\newblock {Gaussian} fluctuation for the number of particles in {Airy}, bessel,
  sine, and other determinantal random point fields.
\newblock \emph{Journal of Statistical Physics}, 100\penalty0 (3-4):\penalty0
  491--522, 2000.
\newblock \doi{10.1023/A:1018672622921}.

\bibitem[Stam(1959)]{Stam1959}
A.~J. Stam.
\newblock Some inequalities satisfied by the quantities of information of
  {Fisher} and shannon.
\newblock \emph{Information and Control}, 2\penalty0 (2):\penalty0 101--112,
  1959.
\newblock \doi{10.1016/S0019-9958(59)90348-1}.

\bibitem[Stam(1982)]{Stam1982}
A.~J. Stam.
\newblock Limit theorems for uniform distributions on spheres in
  high-dimensional {E}uclidean spaces.
\newblock \emph{Journal of Applied Probability}, 19\penalty0 (1):\penalty0
  221--228, 1982.
\newblock \doi{10.2307/3213932}.

\bibitem[Stanley(1968)]{Stanley1968}
H.~E. Stanley.
\newblock Spherical model as the limit of infinite spin dimensionality.
\newblock \emph{Physical Review}, 176\penalty0 (2):\penalty0 718--722, 1968.
\newblock \doi{10.1103/PhysRev.176.718}.

\bibitem[Stein(1972)]{Stein1972}
Charles Stein.
\newblock A bound for the error in the normal approximation to the distribution
  of a sum of dependent random variables.
\newblock In \emph{Proceedings of the Sixth Berkeley Symposium on Mathematical
  Statistics and Probability}, volume~2, pages 583--602, Berkeley, 1972.
  University of California Press.

\bibitem[Subag(2024)]{Subag2024}
Eliran Subag.
\newblock Free energy landscapes in spherical spin glasses.
\newblock \emph{Duke Mathematical Journal}, 173\penalty0 (7):\penalty0
  1291--1357, 2024.
\newblock \doi{10.1215/00127094-2023-0033}.

\bibitem[Talagrand(2006)]{Talagrand2006}
Michel Talagrand.
\newblock Free energy of the spherical mean field model.
\newblock \emph{Probability Theory and Related Fields}, 134:\penalty0 339--382,
  2006.
\newblock \doi{10.1007/s00440-005-0433-8}.

\bibitem[Teyssier(2020)]{Teyssier2020}
Lucas Teyssier.
\newblock Limit profile for random transpositions.
\newblock \emph{The Annals of Probability}, 48\penalty0 (5):\penalty0
  2323--2343, 2020.
\newblock \doi{10.1214/20-AOP1424}.

\bibitem[Tracy and Widom(1996)]{TracyWidom1996}
Craig~A. Tracy and Harold Widom.
\newblock On orthogonal and symplectic matrix ensembles.
\newblock \emph{Communications in Mathematical Physics}, 177\penalty0
  (3):\penalty0 727--754, 1996.
\newblock \doi{10.1007/BF02099545}.

\bibitem[van~der Vaart(1998)]{vanDerVaart1998}
A.~W. van~der Vaart.
\newblock \emph{Asymptotic Statistics}, volume~3 of \emph{Cambridge Series in
  Statistical and Probabilistic Mathematics}.
\newblock Cambridge University Press, 1998.
\newblock \doi{10.1017/CBO9780511802256}.

\bibitem[van Erven and Harremo{\"e}s(2014)]{vanErvenHarremoes2014}
Tim van Erven and Peter Harremo{\"e}s.
\newblock R{\'e}nyi divergence and kullback--leibler divergence.
\newblock \emph{IEEE Transactions on Information Theory}, 60\penalty0
  (7):\penalty0 3797--3820, 2014.
\newblock \doi{10.1109/TIT.2014.2320500}.
\newblock Statement locators refer to v2.

\bibitem[Vempala and Wibisono(2019)]{VempalaWibisono2019}
Santosh~S. Vempala and Andre Wibisono.
\newblock Rapid convergence of the unadjusted {L}angevin algorithm:
  isoperimetry suffices, 2019.
\newblock Version 4, 2 March 2022; statement locators refer to v4.

\bibitem[Voigt(1912)]{Voigt1912}
Woldemar Voigt.
\newblock {\"U}ber das gesetz der intensit{\"a}tsverteilung innerhalb der
  linien eines gasspektrums.
\newblock \emph{Sitzungsberichte der mathematisch-physikalischen Klasse der K.
  B. Akademie der Wissenschaften zu M{\"u}nchen}, pages 603--620, 1912.
\newblock URL \url{https://publikationen.badw.de/de/003395768}.

\bibitem[Wang and Tang(2024)]{WangTang2024}
Ding Wang and Lei-Han Tang.
\newblock Exact finite-size scaling of the maximum likelihood spectra in the
  quenched and annealed {Sherrington-Kirkpatrick} spin glass.
\newblock \emph{SciPost Physics}, 17\penalty0 (4):\penalty0 106, 2024.
\newblock \doi{10.21468/SciPostPhys.17.4.106}.
\newblock Locators refer to arXiv:2405.14215v2.

\bibitem[Widder(1941)]{Widder1941}
David~Vernon Widder.
\newblock \emph{The {Laplace} Transform}, volume~6 of \emph{Princeton
  Mathematical Series}.
\newblock Princeton University Press, Princeton, 1941.

\bibitem[Wigner(1955)]{Wigner1955}
Eugene~P. Wigner.
\newblock Characteristic vectors of bordered matrices with infinite dimensions.
\newblock \emph{Annals of Mathematics}, 62\penalty0 (3):\penalty0 548--564,
  1955.
\newblock \doi{10.2307/1970079}.

\bibitem[Wigner(1958)]{Wigner1958}
Eugene~P. Wigner.
\newblock On the distribution of the roots of certain symmetric matrices.
\newblock \emph{Annals of Mathematics}, 67\penalty0 (2):\penalty0 325--327,
  1958.
\newblock \doi{10.2307/1970008}.

\bibitem[Williams(1991)]{Williams1991}
David Williams.
\newblock \emph{Probability with Martingales}.
\newblock Cambridge Mathematical Textbooks. Cambridge University Press,
  Cambridge, 1991.
\newblock \doi{10.1017/CBO9780511813658}.

\bibitem[Yu(2024)]{Yu2024}
Tingzhou Yu.
\newblock Analyzing dynamics and average case complexity in the spherical
  {Sherrington--Kirkpatrick} model: a focus on extreme eigenvectors, 2024.
\newblock URL \url{https://arxiv.org/abs/2401.03668v1}.
\newblock Version 1, 8 January 2024.

\bibitem[Zippold et~al.(2000)Zippold, K{\"u}hn, and
  Horner]{ZippoldKuhnHorner2000}
W.~Zippold, R.~K{\"u}hn, and H.~Horner.
\newblock Non-equilibrium dynamics of simple spherical spin models.
\newblock \emph{The European Physical Journal B}, 13\penalty0 (3):\penalty0
  531--537, 2000.
\newblock \doi{10.1007/s100510050065}.
\newblock Locators refer to arXiv:cond-mat/9904329v1.

\end{thebibliography}