The Parisi Formula has a Unique Minimizer


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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Corresponding author

Correspondence to Wei-Kuo Chen.

Additional information

The research of A. A. is supported by NSF grant DMS-1407554.

Communicated by F. Toninelli

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Auffinger, A., Chen, WK. The Parisi Formula has a Unique Minimizer. Commun. Math. Phys. 335, 1429–1444 (2015).

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  • Spin Glass
  • Variational Representation
  • Dominate Convergence Theorem
  • Variational Formula
  • Standard Brownian Motion