The Annals of Probability

Parisi formula for the ground state energy in the mixed $p$-spin model

Antonio Auffinger and Wei-Kuo Chen

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Abstract

We show that the thermodynamic limit of the ground state energy in the mixed $p$-spin model can be identified as a variational problem. This gives a natural generalization of the Parisi formula at zero temperature.

Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4617-4631.

Dates
Received: June 2016
Revised: December 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069268

Digital Object Identifier
doi:10.1214/16-AOP1173

Mathematical Reviews number (MathSciNet)
MR3737919

Zentralblatt MATH identifier
06838128

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Spin glasses ground state energy Parisi formula

Citation

Auffinger, Antonio; Chen, Wei-Kuo. Parisi formula for the ground state energy in the mixed $p$-spin model. Ann. Probab. 45 (2017), no. 6B, 4617--4631. doi:10.1214/16-AOP1173. https://projecteuclid.org/euclid.aop/1513069268


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References

  • [1] Auffinger, A. and Chen, W.-K. (2015). On properties of Parisi measures. Probab. Theory Related Fields 161 817–850.
  • [2] Auffinger, A. and Chen, W.-K. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
  • [3] Bovier, A. and Klimovsky, A. (2009). The Aizenman–Sims–Starr and Guerra’s schemes for the SK model with multidimensional spins. Electron. J. Probab. 14 161–241.
  • [4] Chen, W.-K., Handschy, M. and Lerman, G. (2016). On the energy landscape of the mixed even p-spin model. Available at arXiv:1609.04368.
  • [5] Chen, W.-K. and Sen, A. (2015). Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed $p$-spin models. Available at arXiv:1512.08492.
  • [6] Crisanti, A. and Rizzo, T. (2002). Analysis of the $\infty$-replica symmetry breaking solution of the Sherrington–Kirkpatrick model. Phys. Rev. E (3) 65 046137.
  • [7] Dembo, A., Montanari, A. and Sen, S. (2017). Extremal cuts of sparse random graphs. Ann. Probab. 45 1190–1217.
  • [8] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • [9] Jagannath, A. and Tobasco, I. (2016). A dynamic programming approach to the Parisi functional. Proc. Amer. Math. Soc. 144 3135–3150.
  • [10] Jagannath, A. and Tobasco, I. (2016). Low temperature asymptotics of spherical mean field spin glasses. Available at arXiv:1602.00657.
  • [11] Kim, S.-Y., Lee, S. and Lee, J. (2007). Ground-state energy and energy landscape of the Sherrington–Kirkpatrick spin glass. Phys. Rev. B 76 184412.
  • [12] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
  • [13] Oppermann, R., Schmidt, M. J. and Sherrington, D. (2007). Double criticality of the Sherrington–Kirkpatrick model at $T=0$. Phys. Rev. Lett. 98 127201.
  • [14] Oppermann, R. and Sherrington, D. (2005). Scaling and renormalization group in replica-symmetry-breaking space: Evidence for a simple analytical solution of the Sherrington–Kirkpatrick model at zero temperature. Phys. Rev. Lett. 95 197203.
  • [15] Panchenko, D. (2005). Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17 793–857.
  • [16] Panchenko, D. (2008). On differentiability of the Parisi formula. Electron. Commun. Probab. 13 241–247.
  • [17] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
  • [18] Panchenko, D. (2015). The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Probab. 43 3494–3513.
  • [19] Panchenko, D. (2015). Free energy in the Potts spin glass. Available at arXiv:1512.00370.
  • [20] Panchenko, D. (2015). Free energy in the mixed $p$-spin models with vector spins. Available at arXiv:1512.04441.
  • [21] Pankov, S. (2006). Low-temperature solution of the Sherrington–Kirkpatrick model. Phys. Rev. Lett. 96 197204.
  • [22] Parisi, G. (1979). Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43 1754–1756.
  • [23] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [24] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 55. Springer, Berlin.
  • [25] Talagrand, M. and Spin (2003). 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 46. Springer, Berlin.