The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins |
| 2. | Creator | Author's name, affiliation, country | Anton Bovier; Institut fuer Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universitaet |
| 2. | Creator | Author's name, affiliation, country | Anton Klimovsky; Department Mathematik, Friedrich-Alexander-Universitaet Erlangen-Nuernberg |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Sherrington-Kirkpatrick model, multidimensional spins, quenched large deviations, concentration of measure, Gaussian spins, convexity, Parisi functional, Parisi formula |
| 3. | Subject | Subject classification | Primary 60K35, 82B44, 60F10 |
| 4. | Description | Abstract | 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. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | DFG Research Training Group Stochastic Processes and Probabilistic Analysis, Helmholz-Gemeinschaft |
| 7. | Date | (YYYY-MM-DD) | 2009-01-29 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/611 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v14-611 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 14 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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