Abstract
In 1979, Parisi (Phys Rev Lett 43:1754–1756, 1979) predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington–Kirkpatrick model, and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand (Ann Math 163(1):221–263, 2006) and later generalized to the mixed p-spin models by Panchenko (Ann Probab 42(3):946–958, 2014). In this paper, we prove that the minimizer in Parisi’s formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.
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Auffinger, A., Chen, W.-K.: On properties of Parisi Measures. To appear in Probab. Theory Relat. Fields. preprint arXiv:1303.3573 (2013)
Borell, C.: Diffusion equations and geometric inequalities. Potential Anal. 12(1), 49–71 (2000)
Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)
Chen, W.-K.: Partial results on the convexity of the Parisi functional with PDE approach. To appear in Proc. Am. Math. Soc. preprint arXiv:1308.6559 (2013)
Fleming, W.H.: Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4, 329–346 (1978)
Ghatak, S., Sherrington, D.: Crystal field effects in a general S Ising spin glass. J. Phys. C Solid State Phys. 10, 3149 (1977)
Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233(1), 1–12 (2003)
Karatzas, I., Shreve, S.: Brownian motion and stochastic calculus. 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)
Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. World Scientific Lecture Notes in Physics, vol. 9, World Scientific Publishing Co. Inc., Teaneck (1987)
Panchenko, D.: A question about the Parisi functional. Elect. Comm. Prob. 10, 155–166 (2005)
Panchenko, D.: Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17(7), 793–857 (2005)
Panchenko, D.: The Parisi formula for mixed p-spin models. Ann. Probab. 42(3), 946–958 (2014)
Panchenko, D.: The Sherrington–Kirkpatrick model. Springer Monographs in Mathematics. Springer, New York (2013)
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 SK model for spin glasses. J. Phys. A. 13, L115 (1980)
Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792–1796 (1975)
Talagrand, M.: Mean field models for spin glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vols. 54, 55. Springer, New York (2011)
Talagrand, M.: Parisi measures. J. Funct. Anal. 231(2), 269–286 (2006)
Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)
Talagrand, M.: Free energy of the spherical mean field model. Probab. Theor. Related Fields 134(3), 339–382 (2006)
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Communicated by F. Toninelli
The research of A. A. is supported by NSF grant DMS-1407554.
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Auffinger, A., Chen, WK. The Parisi Formula has a Unique Minimizer. Commun. Math. Phys. 335, 1429–1444 (2015). https://doi.org/10.1007/s00220-014-2254-z
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DOI: https://doi.org/10.1007/s00220-014-2254-z