Electronic Journal of Probability

The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins

Anton Bovier and Anton Klimovsky

Full-text: Open access

Abstract

We prove upper and lower bounds on the free energy of the Sherrington-Kirkpatrick model with multidimensional spins in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle's probability cascades. For this purpose, an abstract quenched large deviations principle of the Gärtner-Ellis type is obtained. We derive Talagrand's representation of Guerra's remainder term for the Sherrington-Kirkpatrick model with multidimensional spins. The derivation is based on well-known properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which shed some light on the original convexity problem of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of a priori estimates.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 8, 161-241.

Dates
Accepted: 29 January 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819469

Digital Object Identifier
doi:10.1214/EJP.v14-611

Mathematical Reviews number (MathSciNet)
MR2471664

Zentralblatt MATH identifier
1205.60166

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60F10: Large deviations

Keywords
Sherrington-Kirkpatrick model multidimensional spins quenched large deviations concentration of measure Gaussian spins convexity Parisi functional Parisi formula

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bovier, Anton; Klimovsky, Anton. The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins. Electron. J. Probab. 14 (2009), paper no. 8, 161--241. doi:10.1214/EJP.v14-611. https://projecteuclid.org/euclid.ejp/1464819469


Export citation

References

  • Michael Aizenman, Robert Sims, and Shannon L. Starr. An Extended Variational Principle for the SK Spin-Glass Model. Phys. Rev. B, 68:214403, 2003.
  • Aizenman, Michael; Sims, Robert; Starr, Shannon L. Mean-field spin glass models from the cavity-ROSt perspective. Prospects in mathematical physics, 1–30, Contemp. Math., 437, Amer. Math. Soc., Providence, RI, 2007.
  • Louis-Pierre Arguin. Spin glass computations and Ruelle's probability cascades. J. Stat. Phys., 126(4-5):951–976, 2007. arXiv:math-ph/0608045v1.
  • Ben Arous, G.; Dembo, A.; Guionnet, A. Aging of spherical spin glasses. Probab. Theory Related Fields 120 (2001), no. 1, 1–67.
  • Bogachev, Vladimir I. Gaussian measures.Mathematical Surveys and Monographs, 62. American Mathematical Society, Providence, RI, 1998. xii+433 pp. ISBN: 0-8218-1054-5
  • Erwin Bolthausen and Alain-Sol Sznitman. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys., 197(2):247–276, 1998.
  • Bovier, Anton. Statistical mechanics of disordered systems.A mathematical perspective.Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006. xiv+312 pp. ISBN: 978-0-521-84991-3; 0-521-84991-8
  • Bovier, Anton; Kurkova, Irina. Much ado about Derrida's GREM. Spin glasses, 81–115, Lecture Notes in Math., 1900, Springer, Berlin, 2007.
  • Briand, Philippe; Hu, Ying. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 (2008), no. 3-4, 543–567.
  • Comets, F. Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Theory Related Fields 80 (1989), no. 3, 407–432.
  • Andrea Crisanti and Hans-Jurgen Sommers. The spherical p-spin interaction spin glass model: the statics. Zeitschrift fur Physik B Condensed Matter, 87(3):341–354, 1992.
  • Da Lio, Francesca; Ley, Olivier. Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006), no. 1, 74–106 (electronic).
  • den Hollander, Frank. Large deviations.Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. x+143 pp. ISBN: 0-8218-1989-5
  • Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
  • Fröhlich, J.; Zegarliźski, B. Some comments on the Sherrington-Kirkpatrick model of spin glasses. Comm. Math. Phys. 112 (1987), no. 4, 553–566.
  • Griffiths, Robert B. A proof that the free energy of a spin system is extensive. J. Mathematical Phys. 5 1964 1215–1222.
  • Guerra, Francesco. Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 (2003), no. 1, 1–12.
  • Guerra, Francesco. Mathematical aspects of mean field spin glass theory. European Congress of Mathematics, 719–732, Eur. Math. Soc., Zürich, 2005.
  • Francesco Guerra and Fabio Lucio Toninelli. The infinite volume limit in generalized mean field disordered models. Markov Process. Related Fields, 9(2):195–207, 2003.
  • Panchenko, Dmitry. A question about the Parisi functional. Electron. Comm. Probab. 10 (2005), 155–166 (electronic).
  • Panchenko, Dmitry. Free energy in the generalized Sherrington-Kirkpatrick mean field model. Rev. Math. Phys. 17 (2005), no. 7, 793–857.
  • Panchenko Dmitry and Talagrand Michel. Guerra's interpolation using Derrida-Ruelle cascades. Preprint, 2007. arXiv:0708.3641v2 [math.PR].
  • Panchenko, Dmitry; Talagrand, Michel. On the overlap in the multiple spherical SK models. Ann. Probab. 35 (2007), no. 6, 2321–2355.
  • Ruelle, David. A mathematical reformulation of Derrida's REM and GREM. Comm. Math. Phys. 108 (1987), no. 2, 225–239.
  • Sherrington, David. Spin glasses: a perspective. Spin glasses, 45–62, Lecture Notes in Math., 1900, Springer, Berlin, 2007.
  • David Sherrington and Scott Kirkpatrick. Solvable Model of a Spin-Glass. Physical Review Letters, 35(26):1792–1796, 1975.
  • Talagrand, Michel. Large deviation principles and generalized Sherrington-Kirkpatrick models.Probability theory. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 2, 203–244.
  • Talagrand, Michel. 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-Verlag, Berlin, 2003. x+586 pp. ISBN: 3-540-00356-8
  • Talagrand Michel. Free energy of the spherical mean field model. Probab. Theory Related Fields, 134(3):339–382, 2006.
  • Talagrand, Michel. The Parisi formula. Ann. of Math. (2) 163 (2006), no. 1, 221–263.
  • Talagrand, Michel. Parisi measures. J. Funct. Anal. 231 (2006), no. 2, 269–286.
  • Talagrand, Michel. Large deviations, Guerra's and A.S.S. schemes, and the Parisi hypothesis. J. Stat. Phys. 126 (2007), no. 4-5, 837–894.
  • Talagrand, Michel. Mean field models for spin glasses: some obnoxious problems. Spin glasses, 63–80, Lecture Notes in Math., 1900, Springer, Berlin, 2007.
  • Toubol, Alain. High temperature regime for a multidimensional Sherrington-Kirkpatrick model of spin glass. Probab. Theory Related Fields 110 (1998), no. 4, 497–534.