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The Replica Symmetric Solution for Potts Models on d-Regular Graphs

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Abstract

We establish an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions.

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References

  1. Aizenman M., Sims R., Starr S.L.: Extended variational principle for the Sherrington–Kirkpatrick spin-glass model. Phys. Rev. B 68, 214403 (2003)

    Article  ADS  Google Scholar 

  2. Abou-Chacra R., Thouless D., Anderson P.: A selfconsistent theory of localization. J. Phys. C 6(10), 1734 (1973)

    Article  ADS  Google Scholar 

  3. Aizenman, M., Warzel, S.: The canopy graph and level statistics for random operators on trees. Math. Phys. Anal. Geom. 9(4), 291–333 (2007), 2006

    Google Scholar 

  4. Borgs C., Chayes J., Kahn J., Lovász L.: Left and right convergence of graphs with bounded degree. Rand. Struct. Alg. 42(1), 1–28 (2013)

    Article  MATH  Google Scholar 

  5. Bethe H.A.: Statistical theory of superlattices. Proc. R. Soc. Lond. A 150(871), 552–575 (1935)

    Article  ADS  MATH  Google Scholar 

  6. Bayati, M., Gamarnik, D., Tetali, P.: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In: Proc. ACM Symp. (STOC). New York: ACM, 2010, pp. 105–114

  7. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 1–13 (electronic), (2001)

    Google Scholar 

  8. Chayes J.T., Chayes L., Sethna J.P., Thouless D.J.: A mean field spin glass with short-range interactions. Commun. Math. Phys. 106(1), 41–89 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  9. Contucci P., Dommers S., Giardinà à C., Starr S.: Antiferromagnetic Potts model on the Erdős-Rényi random graph. Commun. Math. Phys. 323(2), 517–554 (2013)

    Article  ADS  MATH  Google Scholar 

  10. Chalupa J., Leath P., Reich G.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12(1), L31 (1979)

    Article  ADS  Google Scholar 

  11. Dommers S., Giardinà à C., Hofstad R.: Ising models on power-law random graphs. J. Stat. Phys. 141(4), 638–660 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Dembo A., Kagan A., Shepp L.A.: Remarks on the maximum correlation coefficient. Bernoulli 7(2), 343–350 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dembo A., Montanari A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24(2), 137–211 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dembo A., Montanari A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dembo A., Montanari A., Sun N.: Factor models on locally tree-like graphs. Ann. Probab. 41(6), 4162–4213 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Guerra F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Janson S., Łuczak T., Rucinski A.: Random graphs Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)

    Google Scholar 

  18. Krza̧kała, F., Montanari, A., Ricci-Tersenghi, F. Semerjian, G., Zdeborová, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA 104(25), 10318–10323 (electronic) (2007)

    Google Scholar 

  19. McDiarmid, C.: Concentration. In: Probabilistic methods for algorithmic discrete mathematics, Vol. 16 of Algorithms Combin. Berlin: Springer, 1998, pp. 195–248

  20. Mézard M., Montanari A.: Information, physics, and computation. Oxford Graduate Texts. Oxford University Press, Oxford (2009)

    Book  Google Scholar 

  21. Montanari A., Mossel E., Sly A.: The weak limit of Ising models on locally tree-like graphs. Probab. Theory Relat. Fields 152(1–2), 31–51 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mézard M., Parisi G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B Condens. Matter Phys. 20(2), 217–233 (2001)

    MathSciNet  Google Scholar 

  23. Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. In: World scientific lecture notes in physics, Vol. 9. Teaneck: World Scientific Publishing Co. Inc., 1987

  24. Mézard M., Parisi G., Zecchina R.: Analytic and algorithmic solution of random satisfiability problems. Science 297(5582), 812–815 (2002)

    Article  ADS  Google Scholar 

  25. Robbins H.: A remark on Stirling’s formula. Am. Math. Monthly 62, 26–29 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  26. Spitzer F.: Markov random fields on an infinite tree. Ann. Probab. 3(3), 387–398 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sly, A., Sun, N.: Counting in two-spin models on d-regular graphs. Ann Probab. (2013) [to appear]

  28. Talagrand, M.: Mean field models for spin glasses. In: Volume I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 54. 3. Folge. Berlin: Springer, Basic examples (2011)

  29. Thouless D.: Spin-glass on a Bethe lattice. Phys. Rev. Lett. 56(10), 1082 (1986)

    Article  ADS  Google Scholar 

  30. Weiss P.R.: The application of the Bethe–Peierls method to ferromagnetism. Phys. Rev. 74(10), 1493 (1948)

    Article  ADS  MATH  Google Scholar 

  31. Zachary S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11(4), 894–903 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Building 380, Sloan Hall, Stanford, CA, 94305, USA

    Amir Dembo

  2. Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, CA, 94305, USA

    Andrea Montanari

  3. Department of Statistics, University of California, Berkeley, Evans Hall, Berkeley, CA, 94720, USA

    Allan Sly

  4. Mathematical Science Institute, Australian National University, John Dedman Building, Union Lane, Canberra, ACT, 0200, Australia

    Allan Sly

  5. Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA, 94305, USA

    Amir Dembo, Andrea Montanari & Nike Sun

Authors
  1. Amir Dembo
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  2. Andrea Montanari
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  3. Allan Sly
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  4. Nike Sun
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Corresponding author

Correspondence to Nike Sun.

Additional information

Communicated by H. Spohn

Research partially supported by NSF grants A. Dembo, A. Montanari, N. Sun: DMS-1106627 and A. Montanari: CCF-0743978, A. Sly: Alfred P. Sloan Research Fellowship, and N. Sun: Department of Defense NDSEG Fellowship.

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Dembo, A., Montanari, A., Sly, A. et al. The Replica Symmetric Solution for Potts Models on d-Regular Graphs. Commun. Math. Phys. 327, 551–575 (2014). https://doi.org/10.1007/s00220-014-1956-6

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  • DOI: https://doi.org/10.1007/s00220-014-1956-6

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