Abstract
Thouless et al. (Phys Mag 35(3):593–601, 1977), derived a representation for the free energy of the Sherrington–Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed p-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards–Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed p-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.
Similar content being viewed by others
References
Anderson, P.W.: Lectures on amorphous systems. In: Les Houches, Session XXXI: Ill-Condensed Matter, pp. 159–261 (1978)
Arguin L.-P., Aizenman M.: On the structure of quasi-stationary competing particle systems. Ann. Probab. 37(3), 1080–1113 (2009)
Auffinger A., Chen W.-K.: On properties of Parisi measures. Probab. Theory Related Fields 161(3–4), 817–850 (2015)
Auffinger A., Chen W.-K.: The Parisi formula has a unique minimizer. Commun. Math. Phys. 335(3), 1429–1444 (2015)
Auffinger, A., Chen, W.-K.: Parisi formula for the ground state energy in the mixed p-spin model. ArXiv e-prints, June (2016)
Auffinger, A., Jagannath, A.: Thouless–Anderson–Palmer equations for conditional Gibbs measures in the generic p-spin glass model. ArXiv e-prints, December (2016)
Austin, T.: Measure concentration and the weak Pinsker property. ArXiv e-prints, (2017)
Bolthausen E.: An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Commun. Math. Phys. 325(1), 333–366 (2014)
Boucheron S., Lugosi G., Massart P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford (2013)
Bray A.J., Moore M.A., Young A.P.: Weighted averages of TAP solutions and Parisi’s q(x). J. Phys. C: Solid State Phys. 17, L155 (1984)
Cavagna A., Giardina I., Parisi G., Mézard M.: On the formal equivalence of the TAP and thermodynamic methods in the SK model. J. Phys. A: Math. Gen. 36, 1175–1194 (2003)
Chatterjee S.: Spin glasses and Stein’s method. Probab. Theory Related Fields 148(3-4), 567–600 (2010)
Chatterjee S., Dembo A.: Nonlinear large deviations. Adv. Math. 299, 396–450 (2016)
Chen, W.-K.: Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound. ArXiv e-prints, January (2015)
Chen, W.-K., Handschy, M., Lerman, G.: On the energy landscape of the mixed even p-spin model. ArXiv e-prints (2016)
De Dominicis C., Young A.P.: Weighted averages and order parameters for the infinite range Ising spin glass. J. Phys. A: Math. Gen. 16, 2063 (1983)
Eldan, R.: Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations. ArXiv e-prints, December (2016)
Eldan, R., Gross, R.: Decomposition of mean-field Gibbs distributions into product measures. ArXiv e-prints, August (2017)
Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. In: Volume 25 of Stochastic Modelling and Applied Probability. Springer, New York, 2nd edn (2006)
Guerra F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)
Jagannath A.: Approximate ultrametricity for random measures and applications to spin glasses. Commun. Pure Appl. Math. 70, 611–664 (2017)
Jagannath, A., Sen, S.: On the unbalanced cut problem and the generalized Sherrington–Kirkpatrick model. ArXiv e-prints, July (2017)
Jagannath A., Jagannath A., Jagannath A.: A dynamic programming approach to the parisi functional. Proc. Am. Math. Soc. 144, 3135–3150 (2016)
Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. In: Volume 9 of World Scientific Lecture Notes in Physics. World Scientific Publishing Co., Inc., Teaneck, NJ (1987)
Opper, M., Saad, D. (eds).: Advanced mean field methods. Neural Information Processing Series. MIT Press, Cambridge, MA, 2001. Theory and practice, Papers from the workshop held at Aston University, Birmingham, (1999), A Bradford Book
Panchenko D.: Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17(7), 793–857 (2005)
Panchenko, D.: The Sherrington–Kirkpatrick model. Springer Monographs in Mathematics. Springer, New York (2013)
Panchenko D.: Spin glass models from the point of view of spin distributions. Ann. Probab. 41(3A), 1315–1361 (2013)
Panchenko D.: The Parisi ultrametricity conjecture. Ann. Math. 177(1), 383–393 (2013)
Panchenko D.: The Parisi formula for mixed p-spin models. Ann. Probab. 42(3), 946–958 (2014)
Panchenko, D.: Free energy in the mixed p-spin models with vector spins. ArXiv e-prints, December (2015)
Panchenko, D.: Free energy in the Potts spin glass. ArXiv e-prints, December (2015)
Parisi G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979)
Parisi G.: A sequence of approximate solutions to the S–K model for spin glasses. J. Phys. A 13, L–115 (1980)
Parisi G.: Order parameter for spin glasses. Phys. Rev. Lett. 50, 1946 (1983)
Plefka T.: Convergence condition of the tap equation fo the infinite-ranged ising spin glass model. J. Phys. A: Math. Gen. 15(6), 1971–1978 (1982)
Sherrington D., Kirkpatrick S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35(26), 1792–1795 (1975)
Talagrand M.: The Parisi formula. Ann. Math. 163(1), 221–263 (2006)
Talagrand, M.: Mean field models for spin glasses. In: Volume I, volume 54 of 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]. Springer, Berlin (2011). Basic examples
Talagrand, M.: Mean field models for spin glasses. In: Volume II, volume 55 of 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]. Springer, Heidelberg, (2011). Advanced replica-symmetry and low temperature
Thouless D.J., Anderson P.W., Palmer R.G.: Solution of ‘solvable model of a spin glass’. Phys. Mag. 35(3), 593–601 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Erdos
Rights and permissions
About this article
Cite this article
Chen, WK., Panchenko, D. On the TAP Free Energy in the Mixed p-Spin Models. Commun. Math. Phys. 362, 219–252 (2018). https://doi.org/10.1007/s00220-018-3143-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3143-7