The Annals of Probability

Spin glass models from the point of view of spin distributions

Dmitry Panchenko

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Abstract

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous–Hoover representation encoded by a function $\sigma : [0,1]^{4}\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington–Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington–Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda–Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.

Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1315-1361.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1367241501

Digital Object Identifier
doi:10.1214/11-AOP696

Mathematical Reviews number (MathSciNet)
MR3098679

Zentralblatt MATH identifier
1281.60081

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Mean-field spin glass models perturbations stability

Citation

Panchenko, Dmitry. Spin glass models from the point of view of spin distributions. Ann. Probab. 41 (2013), no. 3A, 1315--1361. doi:10.1214/11-AOP696. https://projecteuclid.org/euclid.aop/1367241501


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References

  • [1] Aizenman, M., Sims, R. and Starr, S. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B 68 214403.
  • [2] Aldous, D. J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math. 1117 1–198. Springer, Berlin.
  • [3] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080–1113.
  • [4] Arguin, L. P. and Chatterjee, S. (2010). Random overlap structures: Properties and applications to spin glasses. Preprint. Available at arxiv:1011.1823.
  • [5] Austin, T. (2008). On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5 80–145.
  • [6] Baffioni, F. and Rosati, F. (2000). Some exact results on the ultrametric overlap distribution in mean field spin glass models. Eur. Phys. J. B 17 439–447.
  • [7] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.
  • [8] Bovier, A. and Kurkova, I. (2004). Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. Henri Poincaré Probab. Stat. 40 439–480.
  • [9] Chatterjee, S. (2009). The Ghirlanda–Guerra identities without averaging. Preprint. Available at arXiv:0911.4520.
  • [10] De Sanctis, L. (2004). Random multi-overlap structures and cavity fields in diluted spin glasses. J. Stat. Phys. 117 785–799.
  • [11] De Sanctis, L. and Franz, S. (2009). Self-averaging identities for random spin systems. In Spin Glasses: Statics and Dynamics. Progress in Probability 62 123–142. Birkhäuser, Basel.
  • [12] Franz, S. and Leone, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111 535–564.
  • [13] Ghirlanda, S. and Guerra, F. (1998). General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31 9149–9155.
  • [14] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • [15] Hoover, D. N. (1982). Row-column exchangeability and a generalized model for probability. In Exchangeability in Probability and Statistics (Rome, 1981) 281–291. North-Holland, Amsterdam.
  • [16] Kallenberg, O. (1989). On the representation theorem for exchangeable arrays. J. Multivariate Anal. 30 137–154.
  • [17] Panchenko, D. (2005). A note on the free energy of the coupled system in the Sherrington–Kirkpatrick model. Markov Process. Related Fields 11 19–36.
  • [18] Panchenko, D. (2007). A note on Talagrand’s positivity principle. Electron. Commun. Probab. 12 401–410 (electronic).
  • [19] Panchenko, D. (2010). A connection between the Ghirlanda–Guerra identities and ultrametricity. Ann. Probab. 38 327–347.
  • [20] Panchenko, D. (2010). On the Dovbysh–Sudakov representation result. Electron. Commun. Probab. 15 330–338.
  • [21] Panchenko, D. (2010). The Ghirlanda–Guerra identities for mixed $p$-spin model. C. R. Math. Acad. Sci. Paris 348 189–192.
  • [22] Panchenko, D. and Talagrand, M. (2004). Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130 319–336.
  • [23] Panchenko, D. and Talagrand, M. (2007). On one property of Derrida–Ruelle cascades. C. R. Math. Acad. Sci. Paris 345 653–656.
  • [24] Parisi, G. (1980). A sequence of approximate solutions to the S–K model for spin glasses. J. Phys. A 13 L115.
  • [25] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
  • [26] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 46. Springer, Berlin.
  • [27] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [28] Talagrand, M. (2006). Parisi measures. J. Funct. Anal. 231 269–286.
  • [29] Talagrand, M. (2010). Construction of pure states in mean field models for spin glasses. Probab. Theory Related Fields 148 601–643.
  • [30] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I: Basic Examples. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 54. Springer, Berlin.