Communications in Mathematical Physics

, Volume 335, Issue 3, pp 1429–1444 | Cite as

The Parisi Formula has a Unique Minimizer

  • Antonio Auffinger
  • Wei-Kuo Chen


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.


Spin Glass Variational Representation Dominate Convergence Theorem Variational Formula Standard Brownian Motion 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA

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