Variational representations for the parisi functional and the two-dimensional Guerra-Talagrand bound

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The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [Ann. of Math. (2) 163 (2006) 221- 263]. The central argument relied on a dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigation remains elusive. Motivated by Bovier-Klimovsky [Electron. J. Probab. 14 (2009) 161-241] and Auffinger-Chen [Comm. Math. Phys. 335 (2015) 1429-1444] the aim of this paper is to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound in the mixed p-spin models in terms of optimal stochastic control problems. We compute the directional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We demonstrate how our approach provides a simple and efficient control for the GT bound that yields several new results on Talagrand's positivity of the overlap and disorder chaos in Chatterjee [Disorder chaos and multiple valleys in spin glasses. Preprint] and Chen [Ann. Probab. 41 (2013) 3345-3391]. In particular, we provide some examples of the models containing odd p-spin interactions.

Original languageEnglish (US)
Pages (from-to)3929-3966
Number of pages38
JournalAnnals of Probability
Issue number6
StatePublished - Nov 1 2017

Bibliographical note

Funding Information:
Received January 2015; revised November 2015. 1Supported in part by AMS-Simons travel grant, NSF Grant DMS-16-42207 and Hong Kong research grant council GRF 14302515. MSC2010 subject classifications. 60K35, 82B44. Key words and phrases. Chaos in disorder, Parisi formula, replica symmetry breaking, Sherrington–Kirkpatrick model.

Publisher Copyright:
© Institute of Mathematical Statistics, 2017.


  • Chaos in disorder
  • Parisi formula
  • Replica symmetry breaking
  • Sherrington-Kirkpatrick model


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