Electronic Journal of Probability

The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins

Anton Bovier and Anton Klimovsky

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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.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 8, 161-241.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60F10: Large deviations

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

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Bovier, Anton; Klimovsky, Anton. The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins. Electron. J. Probab. 14 (2009), paper no. 8, 161--241. doi:10.1214/EJP.v14-611. https://projecteuclid.org/euclid.ejp/1464819469

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