Marginals of a spherical spin Glass model with correlated disorder. (English) Zbl 1502.82010

Summary: In this paper we prove the weak convergence, in the high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions in a subset of the high-temperature phase. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.


82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)


Full Text: DOI arXiv


[1] Michael Aizenman, Robert Sims, and Shannon L Starr. Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Physical Review B, 68(21):214403, 2003. · Zbl 1175.82033
[2] Greg W Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices. Number 118. Cambridge University Press, 2010. · Zbl 1184.15023
[3] Jean Barbier and Nicolas Macris. The adaptive interpolation method: a simple scheme to prove replica formulas in bayesian inference. Probability theory and related fields, 174(3):1133-1185, 2019. · Zbl 1478.60253
[4] Jean Barbier and Nicolas Macris. The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models. Journal of Physics A: Mathematical and Theoretical, 52(29):294002, 2019. · Zbl 1478.60253
[5] David Belius and Nicola Kistler. The Tap-Plefka variational principle for the spherical SK model. Communications in Mathematical Physics, 367(3):991-1017, 2019. · Zbl 1419.82069
[6] Gérard Ben Arous, Amir Dembo, and Alice Guionnet. Aging of spherical spin glasses. Probability Theory and Related Fields, 120(1):1-67, 2001. · Zbl 0993.60055
[7] Bhaswar B Bhattacharya and Subhabrata Sen. High temperature asymptotics of orthogonal mean-field spin glasses. Journal of Statistical Physics, 162(1):63-80, 2016. · Zbl 1336.82009
[8] Herm Jan Brascamp and Elliott H Lieb. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22(4):366-389, 1976. · Zbl 0334.26009
[9] Wei-Kuo Chen. The Aizenman-Sims-Starr scheme and Parisi formula for mixed \(p\)-spin spherical models. Electronic Journal of Probability, 18:1-14, 2013. · Zbl 1288.60127
[10] Amin Coja-Oghlan, Florent Krzakala, Will Perkins, and Lenka Zdeborová. Information-theoretic thresholds from the cavity method. Advances in Mathematics, 333:694-795, 2018. · Zbl 1397.82013
[11] Andrea Crisanti, Heinz Horner, and Hans-Jurgen Sommers. The spherical \(p\)-spin interaction spin-glass model. Zeitschrift für Physik B Condensed Matter, 92(2):257-271, 1993.
[12] Andrea Crisanti and Hans-Jurgen Sommers. The spherical p-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter, 87(3):341-354, 1992.
[13] David L Donoho, Adel Javanmard, and Andrea Montanari. Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing. IEEE Transactions on Information Theory, 59(11):7434-7464, 2013. · Zbl 1364.94120
[14] Zhou Fan. Approximate message passing algorithms for rotationally invariant matrices. arXiv preprint 2008.11892, 2020.
[15] Zhou Fan and Yihong Wu. The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings. arXiv preprint 2105.02797, 2021.
[16] Laura Foini and Jorge Kurchan. Annealed averages in spin and matrix models. arXiv preprint 2104.04363, 2021.
[17] Marylou Gabrié, Andre Manoel, Clément Luneau, Jean Barbier, Nicolas Macris, Florent Krzakala, and Lenka Zdeborová. Entropy and mutual information in models of deep neural networks. Journal of Statistical Mechanics: Theory and Experiment, 2019(12):124014, 2019. · Zbl 1459.94076
[18] Cedric Gerbelot, Alia Abbara, and Florent Krzakala. Asymptotic errors for teacher-student convex generalized linear models (or: How to prove Kabashima’s replica formula). arXiv preprint 2006.06581, 2020.
[19] Francesco Guerra and Fabio Lucio Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics, 230(1):71-79, 2002. · Zbl 1004.82004
[20] Alice Guionnet and Mylene Maıda. A Fourier view on the r-transform and related asymptotics of spherical integrals. Journal of Functional Analysis, 222(2):435-490, 2005. · Zbl 1065.60023
[21] Yoshiyuki Kabashima. A CDMA multiuser detection algorithm on the basis of belief propagation. Journal of Physics A: Mathematical and General, 36(43):11111, 2003. · Zbl 1081.94509
[22] John M Kosterlitz, David J Thouless, and Raymund C Jones. Spherical model of a spin-glass. Physical Review Letters, 36(20):1217, 1976.
[23] Beatrice Laurent and Pascal Massart. Adaptive estimation of a quadratic functional by model selection. Annals of Statistics, pages 1302-1338, 2000. · Zbl 1105.62328
[24] Junjie Ma, Ji Xu, and Arian Maleki. Impact of the sensing spectrum on signal recovery in generalized linear models. arXiv preprint 2111.03237, 2021. · Zbl 1432.94032
[25] Antoine Maillard, Laura Foini, Alejandro Lage Castellanos, Florent Krzakala, Marc Mézard, and Lenka Zdeborová. High-temperature expansions and message passing algorithms. Journal of Statistical Mechanics: Theory and Experiment, 2019(11):113301, 2019. · Zbl 1459.82094
[26] Marc Mézard and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009. · Zbl 1163.94001
[27] Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, volume 9. World Scientific Publishing Company, 1987.
[28] Dmitry Panchenko. The Sherrington-Kirkpatrick model. Springer Science & Business Media, 2013. · Zbl 1266.82005
[29] Giorgio Parisi and Marc Potters. Mean-field equations for spin models with orthogonal interaction matrices. Journal of Physics A: Mathematical and General, 28(18):5267, 1995. · Zbl 0868.60052
[30] Sundeep Rangan, Philip Schniter, and Alyson K Fletcher. Vector approximate message passing. IEEE Transactions on Information Theory, 65(10):6664-6684, 2019. · Zbl 1432.94036
[31] Firas Rassoul-Agha and Timo Seppäläinen. A course on large deviations with an introduction to Gibbs measures, volume 162. American Mathematical Soc., 2015. · Zbl 1330.60001
[32] Eliran Subag. Free energy landscapes in spherical spin glasses. arXiv preprint 1804.10576, 2018. · Zbl 1500.82016
[33] Takashi Takahashi and Yoshiyuki Kabashima. Macroscopic analysis of vector approximate message passing in a model mismatch setting. In 2020 IEEE International Symposium on Information Theory (ISIT), pages 1403-1408. IEEE, 2020. · Zbl 1505.94023
[34] Koujin Takeda, Shinsuke Uda, and Yoshiyuki Kabashima. Analysis of CDMA systems that are characterized by eigenvalue spectrum. EPL (Europhysics Letters), 76(6):1193, 2006.
[35] Michel Talagrand. Free energy of the spherical mean field model. Probability Theory and Related Fields, 134(3):339-382, 2006. · Zbl 1130.82019
[36] Michel Talagrand. Mean field models for spin glasses: Volume I: Basic examples, volume 54. Springer Science & Business Media, 2010.
[37] Min Yan. Extension of convex function. Journal of Convex Analysis, 21(4):965-987, 2014. · Zbl 1312.26027
[38] Lenka Zdeborová and Florent Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453-552, 2016.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.