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November 2017 Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound
Wei-Kuo Chen
Ann. Probab. 45(6A): 3929-3966 (November 2017). DOI: 10.1214/16-AOP1154


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.


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Wei-Kuo Chen. "Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound." Ann. Probab. 45 (6A) 3929 - 3966, November 2017.


Received: 1 January 2015; Revised: 1 November 2015; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838111
MathSciNet: MR3729619
Digital Object Identifier: 10.1214/16-AOP1154

Primary: 60K35 , 82B44

Keywords: Chaos in disorder , Parisi formula , replica symmetry breaking , Sherrington–Kirkpatrick model

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 6A • November 2017
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