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Irreducible Free Energy Expansion and Overlaps Locking in Mean Field Spin Glasses


Following the works of Guerra, 1995; Aizenmar and Contucci, J. State. Phys. 92 (5–6): 765–783 (1998), we introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda–Guerra relationships (GG) (often called Aizenman–Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are “superimposed” by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations and in a form that simply put in evidence how the complexity of the solution is related to the complexity of the entropy.

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  1. D. Sherrington and S. Kirkpatrick, A solvable model of spin glass. Phys. Rew. Lett. 35:1792–1796 (1975).

    Article  ADS  Google Scholar 

  2. D. Sherrington and S. Kirkpatrick, Infinite ranged models of spin glass. Phys. Rew. B17:4384–4403 (1978).

    ADS  Google Scholar 

  3. M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond World Scientific Publishing (1987).

  4. L. A. Pasteur and M. V. Shcherbina, The absence of self averaging of the order parameter in SK model. J. Stat. Phys. 62:1–19 (1991).

    Article  Google Scholar 

  5. S. Ghirlanda and F. Guerra, General properties of overlap probability distributions in disordered spin systems. J. Phys. A 31:9149–9155 (1998).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. F. Guerra, Sum rules for the free energy in the mean field spin glass model. Field Institute Comm. 30 (2001).

  7. E. Marinari, G. Parisi, J. Ruiz-Lorenzo, and F. Ritort, Numerical evidence for spontaneously broken replica symmetry in 3D spin glasses. arXiv: cond-mat/950836v1.

  8. G. Parisi, On the probabilistic formulation of the replica approach to spin glasses. arXiv: cond-mat/9801081v1

  9. F. Guerra, The cavity method in the mean field spin glass model. Functional representation of the thermodynamic variables. Advances in dynamical system and quantum physics. World Scientific (1995).

  10. F. Guerra, About the cavity fields in mean field spin glass models. arXiv: cond-mat/0307673v1.

  11. M. Aizenman and P. Contucci, On the stability of the quenched state in mean field spin glass models. J. Stat. Phys. 92(5–6):765–783 (1998).

    Google Scholar 

  12. G. Parisi, Stochastic stability Disordered and complex system. AIP Conf. Proc.

  13. M. Talagrand, Spin Glasses: A Challenge for Mathematicians. Springer-Verlag, Berlin (2003).

  14. F. Guerra and F. L. Toninelli, The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230(1):71–79 (2002).

    Google Scholar 

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Correspondence to Adriano Barra.

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Barra, A. Irreducible Free Energy Expansion and Overlaps Locking in Mean Field Spin Glasses. J Stat Phys 123, 601–614 (2006).

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Key Words

  • Cavity field
  • Ghirlanda–Guerra
  • stochastic stability
  • Aizenman-Contucci polynomials