The Aizenman-Sims-Starr and Guerra’s schemes for the SK model with multidimensional spins
Anton Bovier, Institut fuer Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universitaet
Anton Klimovsky, Department Mathematik, Friedrich-Alexander-Universitaet Erlangen-Nuernberg
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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.