## The Annals of Probability

### Free energy in the Potts spin glass

Dmitry Panchenko

#### Abstract

We study the Potts spin glass model, which generalizes the Sherrington–Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product, which includes the Potts spin glass as a special case. As another example of application, one can show that Talagrand’s bound for multiple copies of the mixed $p$-spin model with constrained overlaps is asymptotically sharp. We will consider these problems in the subsequent paper and illustrate the main new idea on the technically more transparent case of the Potts spin glass.

#### Article information

Source
Ann. Probab., Volume 46, Number 2 (2018), 829-864.

Dates
Revised: April 2017
First available in Project Euclid: 9 March 2018

https://projecteuclid.org/euclid.aop/1520586270

Digital Object Identifier
doi:10.1214/17-AOP1193

Mathematical Reviews number (MathSciNet)
MR3773375

Zentralblatt MATH identifier
06864074

#### Citation

Panchenko, Dmitry. Free energy in the Potts spin glass. Ann. Probab. 46 (2018), no. 2, 829--864. doi:10.1214/17-AOP1193. https://projecteuclid.org/euclid.aop/1520586270

#### References

• [1] Aizenman, M., Sims, R. and Starr, S. L. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B. 68 214403.
• [2] Barra, A., Contucci, P., Mingione, E. and Tantari, D. (2015). Multi-species mean field spin glasses. Rigorous results. Ann. Henri Poincaré 16 691–708.
• [3] Caltagirone, F., Parisi, G. and Rizzo, T. (2012). Dynamical critical exponents for the mean-field Potts glass. Phys. Rev. E 85 051504.
• [4] Chen, W.-K. (2013). The Aizenman–Sims–Starr scheme and Parisi formula for mixed $p$-spin spherical models. Electron. J. Probab. 18 no. 94, 14.
• [5] Dembo, A., Montanari, A. and Sen, S. (2017). Extremal cuts of sparse random graphs. Ann. Probab. 45 1190–1217.
• [6] De Santis, E., Parisi, G. and Ritort, F. (1995). On the static and dynamical transition in the mean-field Potts glass. J. Phys. A 28 3025–3041.
• [7] Elderfield, D. and Sherrington, D. (1983). The curious case of the Potts spin glass. J. Phys. C, Solid State Phys. 16 L497.
• [8] Elderfield, D. and Sherrington, D. (1983). Novel non-ergodicity in the Potts spin glass. J. Phys. C, Solid State Phys. 16 L1169.
• [9] Erdős, P., Hajnal, A. and Pach, J. (2000). A Ramsey-type theorem for bipartite graphs. Geombinatorics 10 64–68.
• [10] Franz, S., Parisi, G. and Virasoro, M. A. (1992). Ultrametricity in an inhomogeneous simplest spin glass model. Europhys. Lett. 17 5–9.
• [11] Franz, S., Parisi, G. and Virasoro, M. A. (1993). Free-energy cost for ultrametricity violations in spin glasses. Europhys. Lett. 22 405–411.
• [12] Ghatak, S. K. and Sherrington, D. (1977). Crystal field effects in a general $S$ Ising spin glass. J. Phys. C, Solid State Phys. 10 3149.
• [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] Gross, D. J., Kanter, I. and Sompolinsky, S. (1985). Mean-field theory of the Potts glass. Phys. Rev. Lett. 55 304–307.
• [15] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
• [16] Jagannath, A., Ko, J. and Sen, S. (2017). A connection between MAX $\kappa$-CUT and the inhomogeneous Potts spin glass in the large degree limit. Available at arXiv:1703.03455.
• [17] Marinari, E., Mossa, S. and Parisi, G. (1999). Glassy Potts model: A disordered Potts model without a ferromagnetic phase. Phys. Rev. B 59 8401.
• [18] Mézard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. 9. World Scientific, Teaneck, NJ.
• [19] Nishimori, H. and Stephen, M. J. (1983). Gauge-invariant frustrated Potts spin-glass. Phys. Rev. B (3) 27 5644–5652.
• [20] 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.
• [21] Panchenko, D. (2005). Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17 793–857.
• [22] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
• [23] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
• [24] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
• [25] Panchenko, D. (2015). Free energy in the mixed $p$-spin models with vector spins. Preprint. Available at arXiv:1512.04441.
• [26] Panchenko, D. (2015). The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Probab. 43 3494–3513.
• [27] Parisi, G. (1980). A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13 L–115.
• [28] Parisi, G. (1983). Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43 1754–1756.
• [29] Parisi, G. and Talagrand, M. (2004). On the distribution of the overlaps at given disorder. C. R. Math. Acad. Sci. Paris 339 303–306.
• [30] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series, No. 28. Princeton Univ. Press, Princeton, NJ.
• [31] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
• [32] Sen, S. (2016). Optimization on sparse random hypergraphs and spin glasses. Preprint. Available at arXiv:1606.02365.
• [33] Sherrington, D. (2010). Physics and complexity. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 368 1175–1189.
• [34] Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792–1796.
• [35] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
• [36] Talagrand, M. (2006). Free energy of the spherical mean field model. Probab. Theory Related Fields 134 339–382.
• [37] Talagrand, M. (2006). Parisi measures. J. Funct. Anal. 231 269–286.
• [38] Talagrand, M. (2007). Mean field models for spin glasses: Some obnoxious problems. In Spin Glasses. Lecture Notes in Math. 1900 63–80. Springer, Berlin.
• [39] Talagrand, M. (2011). Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 54, 55. Springer, Berlin.