HEXAGONbeta

Aging in Spherical Spin Glass Dynamics

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physics

Contributed by P. M. Aronow ↗, Patrick Lopatto ↗

Version 1 / Oct 11, 2026 / CC BY 4.0

Abstract

We study Langevin dynamics for the pure spherical pp-spin glass with p≥3p\ge3, with covariance NqpNq^p and generator ΔS+β∇SHN⋅∇S\Delta_S+\beta\nabla_SH_N\cdot\nabla_S, started from a uniform point independent of the disorder. We let the dimension tend to infinity on fixed time horizons and study the long-time behavior of the resulting hard-spherical Cugliandolo–Kurchan equations for the correlation CC and the response RR. At every positive temperature, their solution has a unique stationary limit at bounded time differences, which satisfies the fluctuation–dissipation theorem (FDT), and its energy converges. For β≥βd\beta\ge\beta_{\mathrm d}, where βd\beta_{\mathrm d} is the dynamical transition, we show that the plateau qq of the stationary correlation is the larger solution of the marginality relation β2p(p−1)qp−2(1−q)2=1\beta^2p(p-1)q^{p-2}(1-q)^2=1, and we compute the limiting energy. Correlations with every fixed earlier state tend to zero. The limits of large waiting time and large time difference therefore do not commute, which proves aging of the correlation, including at the transition point. We also prove that the response at a late time, as a function of the time difference, converges to its stationary profile in L2L^2 and uniformly over all earlier times; that FDT in the later time holds uniformly over all earlier times; and that the integral of the squared violation of FDT in the earlier time tends to zero, while for β>βd\beta>\beta_{\mathrm d} the positive part (∂sC−R)+(\partial_sC-R)_+ of this violation keeps a positive integrated mass. For β≥βd\beta\ge\beta_{\mathrm d}, the stationary correlation approaches its plateau as a power law, with the exponent predicted by mode-coupling theory and a finite positive amplitude. For the cubic model and β≥βd\beta\ge\beta_{\mathrm d}, including the transition point, we prove a lower bound on the total susceptibility and a product inequality for correlations at three times; for β>βd\beta>\beta_{\mathrm d}, we prove, for each fixed level below the plateau, an explicit asymptotic lower bound on the first passage time to that level multiplied by the entropy production during the passage. The aging proofs for the cubic and quartic models represent the correlation by vectors in a Hilbert space indexed by trees, and general pure interactions are treated with averages over finitely many independent futures and established algorithmic threshold bounds. The full fluctuation–dissipation relation of the aging regime and the time scale of aging remain open.

Provenance statement

Proof discovery and verification were done with a combination of GPT 6 Astra and GPT 6.1 Sol. The writing was done by Opus 5.5 and GPT 6.1 Sol.

Formalizations

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion 6
OpenAI
ChatGPTVersion 6.1

References

  1. Agarwal, Girish S. Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications. Zeitschrift für Physik, vol. 252, no. 1, pp. 25–38. 1972DOI
  2. Altieri, Ada; Biroli, Giulio; Cammarota, Chiara. Dynamical Mean-Field Theory and Aging Dynamics. Journal of Physics A: Mathematical and Theoretical, vol. 53, pp. 375006. 2020arXivDOI
  3. Alexey Andreanov; Alexandre Lefèvre. Crossover from stationary to aging regime in glassy dynamics. Europhysics Letters, vol. 76, no. 5, pp. 919–925. 2006arXivDOI
  4. Aronszajn, Nachman. Theory of reproducing kernels. Transactions of the American Mathematical Society, vol. 68, no. 3, pp. 337–404. 1950DOI
  5. Auffinger, Antonio; Ben Arous, Gérard. Complexity of random smooth functions on the high-dimensional sphere. The Annals of Probability, vol. 41, no. 6, pp. 4214–4247. 2013arXivDOI
  6. Auffinger, Antonio; Chen, Wei-Kuo. On properties of Parisi measures. Probability Theory and Related Fields, vol. 161, no. 3–4, pp. 817–850. 2015DOI
  7. Auffinger, Antonio; Ben Arous, Gérard; Černý, Jiří. Random matrices and complexity of spin glasses. Communications on Pure and Applied Mathematics, vol. 66, no. 2, pp. 165–201. 2013arXivDOI
  8. Baiesi, Marco; Maes, Christian; Wynants, Bram. Fluctuations and response of nonequilibrium states. Physical Review Letters, vol. 103, no. 1, pp. 010602. 2009arXivDOI
  9. Bakry, Dominique; Émery, Michel. Diffusions hypercontractives. Séminaire de Probabilités XIX 1983/84, vol. 1123, pp. 177–206. 1985DOI
  10. Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer, vol. 348. 2014DOI
Show all 122 referencesShow fewer references
  1. Barrat, Alain; Burioni, Raffaella; Mézard, Marc. Ageing classification in glassy dynamics. Journal of Physics A: Mathematical and General, vol. 29, no. 7, pp. 1311–1330. 1996arXivDOI
  2. Barrat, Alain; Franz, Silvio; Parisi, Giorgio. Temperature evolution and bifurcations of metastable states in mean-field spin glasses, with connections with structural glasses. Journal of Physics A: Mathematical and General, vol. 30, no. 16, pp. 5593–5612. 1997arXivDOI
  3. Bayati, Mohsen; Montanari, Andrea. The dynamics of message passing on dense graphs, with applications to compressed sensing. IEEE Transactions on Information Theory, vol. 57, no. 2, pp. 764–785. 2011arXivDOI
  4. Ben Arous, Gérard. Aging and Spin-glass Dynamics. Proceedings of the International Congress of Mathematicians, Beijing 2002, vol. III, pp. 3–14. 2002arXiv
  5. Ben Arous, Gérard; Černý, Jiří. Dynamics of Trap Models. Mathematical Statistical Physics, vol. 83, pp. 331–394. 2006arXivDOI
  6. Ben Arous, Gérard; Černý, Jiří. The arcsine law as a universal aging scheme for trap models. Communications on Pure and Applied Mathematics, vol. 61, no. 3, pp. 289–329. 2008arXivDOI
  7. Ben Arous, Gérard; Guionnet, Alice. Large deviations for Langevin spin glass dynamics. Probability Theory and Related Fields, vol. 102, no. 4, pp. 455–509. 1995DOI
  8. Ben Arous, Gérard; Guionnet, Alice. Symmetric Langevin spin glass dynamics. The Annals of Probability, vol. 25, no. 3, pp. 1367–1422. 1997DOI
  9. Ben Arous, Gérard; Gün, Onur. Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Communications on Pure and Applied Mathematics, vol. 65, no. 1, pp. 77–127. 2012arXivDOI
  10. Ben Arous, Gérard; Jagannath, Aukosh. Shattering versus metastability in spin glasses. Communications on Pure and Applied Mathematics, vol. 77, no. 1, pp. 139–176. 2024arXivDOI
  11. Ben Arous, Gérard; Dembo, Amir; Guionnet, Alice. Aging of spherical spin glasses. Probability Theory and Related Fields, vol. 120, no. 1, pp. 1–67. 2001DOI
  12. Ben Arous, Gérard; Bovier, Anton; Gayrard, Véronique. Glauber Dynamics of the Random Energy Model. I. Metastable Motion on the Extreme States. Communications in Mathematical Physics, vol. 235, no. 3, pp. 379–425. 2003DOI
  13. Ben Arous, Gérard; Bovier, Anton; Gayrard, Véronique. Glauber Dynamics of the Random Energy Model. II. Aging Below the Critical Temperature. Communications in Mathematical Physics, vol. 236, no. 1, pp. 1–54. 2003DOI
  14. Ben Arous, Gérard; Dembo, Amir; Guionnet, Alice. Cugliandolo–Kurchan equations for dynamics of spin-glasses. Probability Theory and Related Fields, vol. 136, no. 4, pp. 619–660. 2006arXivDOI
  15. Ben Arous, Gérard; Bovier, Anton; Černý, Jiří. Universality of the REM for Dynamics of Mean-Field Spin Glasses. Communications in Mathematical Physics, vol. 282, no. 3, pp. 663–695. 2008arXivDOI
  16. Ben Arous, Gérard; Gheissari, Reza; Jagannath, Aukosh. Bounding Flows for Spherical Spin Glass Dynamics. Communications in Mathematical Physics, vol. 373, no. 3, pp. 1011–1048. 2020arXivDOI
  17. Bengtzelius, U.; Götze, W.; Sjölander, A. Dynamics of supercooled liquids and the glass transition. Journal of Physics C: Solid State Physics, vol. 17, pp. 5915–5934. 1984DOI
  18. Berg, Christian; Christensen, Jens Peter Reus; Ressel, Paul. Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Springer, vol. 100. 1984DOI
  19. Billingsley, Patrick. Probability and Measure. John Wiley & Sons. 1995link
  20. Bingham, Nicholas H.; Goldie, Charles M.; Teugels, Jozef L. Regular Variation. Cambridge University Press, vol. 27. 1987DOI
  21. Biroli, Giulio. Dynamical TAP approach to mean field glassy systems. Journal of Physics A: Mathematical and General, vol. 32, no. 48, pp. 8365–8388. 1999arXivDOI
  22. Bolthausen, Erwin. An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Communications in Mathematical Physics, vol. 325, no. 1, pp. 333–366. 2014arXivDOI
  23. Bouchaud, Jean-Philippe. Weak ergodicity breaking and aging in disordered systems. Journal de Physique I, vol. 2, no. 9, pp. 1705–1713. 1992DOI
  24. Bouchaud, Jean-Philippe; Cugliandolo, Leticia F.; Kurchan, Jorge; Mézard, Marc. Mode-coupling approximations, glass theory and disordered systems. Physica A: Statistical Mechanics and its Applications, vol. 226, no. 3–4, pp. 243–273. 1996arXivDOI
  25. Jean-Philippe Bouchaud; Leticia F. Cugliandolo; Jorge Kurchan; Marc Mézard. Out of equilibrium dynamics in spin-glasses and other glassy systems. Spin Glasses and Random Fields, pp. 161–223. 1998arXivDOI
  26. Bovier, Anton; Gayrard, Véronique. Convergence of clock processes in random environments and ageing in the pp-spin SK model. The Annals of Probability, vol. 41, no. 2, pp. 817–847. 2013arXivDOI
  27. Cameron, R. H.; Martin, W. T. Transformations of Wiener Integrals under Translations. Annals of Mathematics, vol. 45, no. 2, pp. 386–396. 1944DOI
  28. Cavagna, Andrea; Giardina, Irene; Parisi, Giorgio. Stationary points of the Thouless–Anderson–Palmer free energy. Physical Review B, vol. 57, pp. 11251–11257. 1998arXivDOI
  29. Celentano, Michael; Cheng, Chen; Montanari, Andrea. The high-dimensional asymptotics of first order methods with random data. 2021. arXiv:2112.07572, version 3, April 2026arXiv
  30. Černý, Jiří; Wassmer, Tobias. Aging of the Metropolis dynamics on the random energy model. Probability Theory and Related Fields, vol. 167, no. 1–2, pp. 253–303. 2017arXivDOI
  31. Chamon, Claudio; Kennett, Malcolm P.; Castillo, Horacio E.; Cugliandolo, Leticia F. Separation of time scales and reparametrization invariance for aging systems. Physical Review Letters, vol. 89, pp. 217201. 2002arXivDOI
  32. Chen, Wei-Kuo. The Aizenman–Sims–Starr scheme and Parisi formula for mixed pp-spin spherical models. Electronic Journal of Probability, vol. 18, pp. no. 94, 1–14. 2013DOI
  33. Chen, Wei-Kuo; Sen, Arnab. Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed pp-spin models. Communications in Mathematical Physics, vol. 350, no. 1, pp. 129–173. 2017DOI
  34. Cipolloni, Riccardo; Cugliandolo, Leticia F. Coupling spherical pp-spin systems. 2026. arXiv:2609.17522, version 1, September 2026arXiv
  35. Citro, Vincenzo; Ricci-Tersenghi, Federico. Strong Ergodicity Breaking in Dynamical Mean-Field Equations for Mixed pp-Spin Glasses. Physical Review Letters, vol. 135, no. 24, pp. 247102. 2025arXivDOI
  36. Crisanti, Andrea; Sommers, Hans-Jürgen. The spherical pp-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter, vol. 87, no. 3, pp. 341–354. 1992DOI
  37. Andrea Crisanti; Hans-Jürgen Sommers. Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model. Journal de Physique I, vol. 5, no. 7, pp. 805–813. 1995arXivDOI
  38. Crisanti, Andrea; Horner, Heinz; Sommers, Hans-Jürgen. The spherical pp-spin interaction spin-glass model: the dynamics. Zeitschrift für Physik B Condensed Matter, vol. 92, no. 2, pp. 257–271. 1993DOI
  39. Cugliandolo, Leticia F. Course 7: Dynamics of Glassy Systems. Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, vol. 77, pp. 367–521. 2004arXivDOI
  40. Leticia F. Cugliandolo; David S. Dean. Full dynamical solution for a spherical spin-glass model. Journal of Physics A: Mathematical and General, vol. 28, no. 15, pp. 4213–4234. 1995arXivDOI
  41. Cugliandolo, Leticia F.; Kurchan, Jorge. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Physical Review Letters, vol. 71, no. 1, pp. 173–176. 1993arXivDOI
  42. Leticia F. Cugliandolo; Jorge Kurchan. On the out-of-equilibrium relaxation of the Sherrington-Kirkpatrick model. Journal of Physics A: Mathematical and General, vol. 27, no. 17, pp. 5749–5772. 1994arXivDOI
  43. Cugliandolo, Leticia F.; Le Doussal, Pierre. Large time nonequilibrium dynamics of a particle in a random potential. Physical Review E, vol. 53, no. 2, pp. 1525–1552. 1996arXivDOI
  44. Cugliandolo, Leticia F.; Dean, David S.; Kurchan, Jorge. Fluctuation-dissipation theorems and entropy production in relaxational systems. Physical Review Letters, vol. 79, no. 12, pp. 2168–2171. 1997arXivDOI
  45. Cugliandolo, Leticia F.; Kurchan, Jorge; Peliti, Luca. Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics. Physical Review E, vol. 55, no. 4, pp. 3898–3914. 1997arXivDOI
  46. Dandi, Yatin; Gamarnik, David; Pernice, Francisco; Zdeborová, Lenka. Rigorous Asymptotics for First-Order Algorithms Through the Dynamical Cavity Method. Proceedings of the Thirty Ninth Conference on Learning Theory, vol. 336, pp. 1611–1646. 2026arXiv
  47. Dembo, Amir; Gheissari, Reza. Diffusions interacting through a random matrix: universality via stochastic Taylor expansion. Probability Theory and Related Fields, vol. 180, no. 3–4, pp. 1057–1097. 2021arXivDOI
  48. Dembo, Amir; Subag, Eliran. Dynamics for spherical spin glasses: disorder dependent initial conditions. Journal of Statistical Physics, vol. 181, no. 2, pp. 465–514. 2020arXivDOI
  49. Dembo, Amir; Subag, Eliran. Dynamics for spherical spin glasses: Gibbs distributed initial conditions. 2025. arXiv version 3, 8 July 2026arXivDOI
  50. Dembo, Amir; Guionnet, Alice; Mazza, Christian. Limiting dynamics for spherical models of spin glasses at high temperature. Journal of Statistical Physics, vol. 128, no. 4, pp. 847–881. 2007arXivDOI
  51. Dembo, Amir; Lubetzky, Eyal; Zeitouni, Ofer. Universality for Langevin-like spin glass dynamics. The Annals of Applied Probability, vol. 31, no. 6, pp. 2864–2880. 2021arXivDOI
  52. Derrida, Bernard. Random-Energy Model: Limit of a Family of Disordered Models. Physical Review Letters, vol. 45, pp. 79–82. 1980DOI
  53. El Alaoui, Ahmed; Montanari, Andrea; Sellke, Mark. Optimization of mean-field spin glasses. The Annals of Probability, vol. 49, no. 6, pp. 2922–2960. 2021arXivDOI
  54. El Alaoui, Ahmed; Montanari, Andrea; Sellke, Mark. Shattering in pure spherical spin glasses. Communications in Mathematical Physics, vol. 406, no. 5, pp. 111. 2025arXivDOI
  55. Fan, Zhou; Ko, Justin; Loureiro, Bruno; Lu, Yue M.; Shen, Yandi. Dynamical mean-field analysis of adaptive Langevin diffusions: Propagation-of-chaos and convergence of the linear response. 2025. arXiv:2504.15556, version 2, November 2025arXiv
  56. Fan, Zhou; Ko, Justin; Loureiro, Bruno; Lu, Yue M.; Shen, Yandi. Dynamical mean-field analysis of adaptive Langevin diffusions: Replica-symmetric fixed point and empirical Bayes. 2025. arXiv:2504.15558, version 2, November 2025arXiv
  57. Fan, Zhou; Misiakiewicz, Theodor; Wang, Leda; Wen, Garrett G. Dynamical mean-field limit and replica-symmetric free energy for the orthogonally-invariant SK model. 2026. arXiv:2607.10102, version 2, September 2026arXiv
  58. Feller, William. An Introduction to Probability Theory and Its Applications. John Wiley & Sons, vol. II. 1971
  59. Folena, Giampaolo; Zamponi, Francesco. On weak ergodicity breaking in mean-field spin glasses. SciPost Physics, vol. 15, no. 3, pp. 109. 2023arXivDOI
  60. Folena, Giampaolo; Franz, Silvio; Ricci-Tersenghi, Federico. Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed pp-Spin Model. Physical Review X, vol. 10, pp. 031045. 2020arXivDOI
  61. Franosch, Thomas; Voigtmann, Thomas. Completely monotone solutions of the mode-coupling theory for mixtures. Journal of Statistical Physics, vol. 109, no. 1–2, pp. 237–259. 2002arXivDOI
  62. Franz, Silvio; Virasoro, Miguel A. Quasi-equilibrium interpretation of ageing dynamics. Journal of Physics A: Mathematical and General, vol. 33, no. 5, pp. 891–905. 2000arXivDOI
  63. Franz, Silvio; Mézard, Marc; Parisi, Giorgio; Peliti, Luca. Measuring Equilibrium Properties in Aging Systems. Physical Review Letters, vol. 81, pp. 1758–1761. 1998arXivDOI
  64. Gardner, Elizabeth. Spin glasses with pp-spin interactions. Nuclear Physics B, vol. 257, pp. 747–765. 1985DOI
  65. Gayrard, Véronique. Aging in Metropolis dynamics of the REM: a proof. Probability Theory and Related Fields, vol. 174, no. 1–2, pp. 501–551. 2019arXivDOI
  66. Gerbelot, Cédric; Troiani, Emanuele; Mignacco, Francesca; Krzakala, Florent; Zdeborová, Lenka. Rigorous Dynamical Mean-Field Theory for Stochastic Gradient Descent Methods. SIAM Journal on Mathematics of Data Science, vol. 6, no. 2, pp. 400–427. 2024arXivDOI
  67. Gheissari, Reza; Jagannath, Aukosh. On the Spectral Gap of Spherical Spin Glass Dynamics. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol. 55, no. 2, pp. 756–776. 2019arXivDOI
  68. Ghimenti, Federico; Berthier, Ludovic; Kurchan, Jorge; van Wijland, Frédéric. Clever algorithms for glasses work by time reparameterization. Proceedings of the National Academy of Sciences, vol. 123, no. 4, pp. e2520818123. 2026. Correction: \em Proc. Natl. Acad. Sci. USA 123(12), e2606723123 (2026), doi:10.1073/pnas.2606723123DOI
  69. Götze, Wolfgang. Some aspects of phase transitions described by the self consistent current relaxation theory. Zeitschrift für Physik B Condensed Matter, vol. 56, pp. 139–154. 1984DOI
  70. Götze, Wolfgang. Properties of the glass instability treated within a mode coupling theory. Zeitschrift für Physik B Condensed Matter, vol. 60, pp. 195–203. 1985DOI
  71. Götze, Wolfgang. The essentials of the mode-coupling theory for glassy dynamics. Condensed Matter Physics, vol. 1, no. 4, pp. 873–904. 1998DOI
  72. Götze, Wolfgang. Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory. Oxford University Press, vol. 143. 2009DOI
  73. Götze, Wolfgang; Sjögren, Lennart. Relaxation processes in supercooled liquids. Reports on Progress in Physics, vol. 55, no. 3, pp. 241–376. 1992DOI
  74. Götze, Wolfgang; Sjögren, Lennart. General properties of certain non-linear integro-differential equations. Journal of Mathematical Analysis and Applications, vol. 195, no. 1, pp. 230–250. 1995DOI
  75. Gross, David J.; Mézard, Marc. The simplest spin glass. Nuclear Physics B, vol. 240, no. 4, pp. 431–452. 1984DOI
  76. Grunwald, Malte. Sanov results for Glauber spin-glass dynamics. Probability Theory and Related Fields, vol. 106, no. 2, pp. 187–232. 1996DOI
  77. Guionnet, Alice. Averaged and quenched propagation of chaos for spin glass dynamics. Probability Theory and Related Fields, vol. 109, no. 2, pp. 183–215. 1997DOI
  78. Guionnet, Alice. Dynamics for Spherical Models of Spin-Glass and Aging. Spin Glasses, vol. 1900, pp. 117–144. 2007DOI
  79. Guionnet, Alice; Mazza, Christian. Long time behaviour of the solution to non-linear Kraichnan equations. Probability Theory and Related Fields, vol. 131, no. 4, pp. 493–518. 2005arXivDOI
  80. Hairer, Martin. An Introduction to Stochastic PDEs. 2025. Version of November 10, 2025link
  81. Higham, Desmond J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, vol. 43, no. 3, pp. 525–546. 2001DOI
  82. Huang, Brice; Sellke, Mark. Algorithmic threshold for multi-species spherical spin glasses. 2023. arXiv version 2, 13 September 2023arXivDOI
  83. Huang, Brice; Sellke, Mark. Optimization Algorithms for Multi-species Spherical Spin Glasses. Journal of Statistical Physics, vol. 191, pp. 29. 2024arXivDOI
  84. Huang, Brice; Sellke, Mark. Tight Lipschitz hardness for optimizing mean field spin glasses. Communications on Pure and Applied Mathematics, vol. 78, no. 1, pp. 60–119. 2025arXivDOI
  85. Janson, Svante. Gaussian Hilbert Spaces. Cambridge University Press, vol. 129. 1997DOI
  86. Bongsoo Kim; Arnulf Latz. The dynamics of the spherical p-spin model: from microscopic to asymptotic. Europhysics Letters, vol. 53, no. 5, pp. 660–666. 2001arXivDOI
  87. Kirkpatrick, T. R.; Thirumalai, Devarajan. Dynamics of the Structural Glass Transition and the pp-Spin-Interaction Spin-Glass Model. Physical Review Letters, vol. 58, pp. 2091–2094. 1987DOI
  88. Kirkpatrick, T. R.; Thirumalai, Devarajan. pp-spin-interaction spin-glass models: Connections with the structural glass problem. Physical Review B, vol. 36, pp. 5388–5397. 1987DOI
  89. Kirkpatrick, T. R.; Thirumalai, Devarajan; Wolynes, Peter G. Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Physical Review A, vol. 40, pp. 1045–1054. 1989DOI
  90. Kubo, Ryogo. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. Journal of the Physical Society of Japan, vol. 12, no. 6, pp. 570–586. 1957DOI
  91. Kunita, Hiroshi. Stochastic Flows and Stochastic Partial Differential Equations. Proceedings of the International Congress of Mathematicians, Berkeley, 1986, vol. 2, pp. 1021–1031. 1987link
  92. Kunita, Hiroshi. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, vol. 24. 1990
  93. Kurchan, Jorge; Laloux, Laurent. Phase space geometry and slow dynamics. Journal of Physics A: Mathematical and General, vol. 29, no. 9, pp. 1929–1948. 1996arXivDOI
  94. Kurchan, Jorge; Parisi, Giorgio; Virasoro, Miguel Angel. Barriers and metastable states as saddle points in the replica approach. Journal de Physique I, vol. 3, no. 8, pp. 1819–1838. 1993DOI
  95. Lang, Johannes; Sachdev, Subir; Diehl, Sebastian. Numerical Renormalization of Glassy Dynamics. Physical Review Letters, vol. 135, pp. 247101. 2025arXivDOI
  96. Lang, Johannes; Citro, Vincenzo; Leuzzi, Luca; Ricci-Tersenghi, Federico. Aging Phase Diagram and Exact Asymptotic Energies of Mixed Spherical Spin Glasses. 2026. arXiv:2608.20623, version 1, August 2026arXiv
  97. Leutheusser, E. Dynamical model of the liquid-glass transition. Physical Review A, vol. 29, no. 5, pp. 2765–2773. 1984DOI
  98. Liang, Tengyuan; Sen, Subhabrata; Sur, Pragya. High-dimensional asymptotics of Langevin dynamics in spiked matrix models. Information and Inference, vol. 12, no. 4, pp. 2720–2752. 2023arXivDOI
  99. Lundgren, Lars; Svedlindh, Per; Nordblad, Per; Beckman, Olov. Dynamics of the Relaxation-Time Spectrum in a CuMn Spin-Glass. Physical Review Letters, vol. 51, pp. 911–914. 1983DOI
  100. Montanari, Andrea. Optimization of the Sherrington–Kirkpatrick Hamiltonian. 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, pp. 1417–1433. 2019arXivDOI
  101. Schur, Issai. Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. Journal für die reine und angewandte Mathematik, vol. 140, pp. 1–28. 1911DOI
  102. Sellke, Mark. The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses. Communications on Pure and Applied Mathematics, vol. 77, no. 11, pp. 4065–4099. 2024arXivDOI
  103. Sellke, Mark. On marginal stability in low temperature spherical spin glasses. Communications in Mathematical Physics, vol. 406, no. 9, pp. 227. 2025. arXiv version 4, 23 August 2026arXivDOI
  104. Sjögren, Lennart. Regenerative processes in supercooled liquids and glasses. Physica A: Statistical Mechanics and its Applications, vol. 322, pp. 81–117. 2003DOI
  105. Sompolinsky, Haim; Zippelius, Annette. Dynamic Theory of the Spin-Glass Phase. Physical Review Letters, vol. 47, pp. 359–362. 1981DOI
  106. Sompolinsky, Haim; Zippelius, Annette. Relaxational dynamics of the Edwards–Anderson model and the mean-field theory of spin-glasses. Physical Review B, vol. 25, pp. 6860–6875. 1982DOI
  107. Subag, Eliran. The complexity of spherical pp-spin models–a second moment approach. The Annals of Probability, vol. 45, no. 5, pp. 3385–3450. 2017arXivDOI
  108. Subag, Eliran. Following the Ground States of Full-RSB Spherical Spin Glasses. Communications on Pure and Applied Mathematics, vol. 74, no. 5, pp. 1021–1044. 2021arXivDOI
  109. Subag, Eliran. The free energy of spherical pure pp-spin models: computation from the TAP approach. Probability Theory and Related Fields, vol. 186, no. 3–4, pp. 715–734. 2023arXivDOI
  110. Talagrand, Michel. Free energy of the spherical mean field model. Probability Theory and Related Fields, vol. 134, no. 3, pp. 339–382. 2006DOI
  111. Vincent, Eric; Hammann, Jacques; Ocio, Miguel; Bouchaud, Jean-Philippe; Cugliandolo, Leticia F. Slow dynamics and aging in spin glasses. Complex Behaviour of Glassy Systems, vol. 492, pp. 184–219. 1997arXivDOI
  112. Widder, David Vernon. The Laplace Transform. Princeton University Press, vol. 6. 1941link

Version history

  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 11, 2026