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2009 The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins
Anton Bovier, Anton Klimovsky
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Electron. J. Probab. 14: 161-241 (2009). DOI: 10.1214/EJP.v14-611


We prove upper and lower bounds on the free energy of the Sherrington-Kirkpatrick model with multidimensional spins in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle's probability cascades. For this purpose, an abstract quenched large deviations principle of the Gärtner-Ellis type is obtained. We derive Talagrand's representation of Guerra's remainder term for the Sherrington-Kirkpatrick model with multidimensional spins. The derivation is based on well-known properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which shed some light on the original convexity problem of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of a priori estimates.


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Anton Bovier. Anton Klimovsky. "The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins." Electron. J. Probab. 14 161 - 241, 2009.


Accepted: 29 January 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1205.60166
MathSciNet: MR2471664
Digital Object Identifier: 10.1214/EJP.v14-611

Primary: 60K35
Secondary: 60F10 , 82B44

Keywords: concentration of measure , convexity , Gaussian spins , multidimensional spins , Parisi formula , Parisi functional , quenched large deviations , Sherrington-Kirkpatrick model

Vol.14 • 2009
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