The Aizenman-Sims-Starr and Guerra's schemes for the SK model with multidimensional spins

A. Bovier, A. Klymovskiy

    Research output: Book/ReportReportAcademic

    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 G¨artner-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.
    Original languageEnglish
    Publishers.n.
    Number of pages60
    Publication statusPublished - 2008

    Publication series

    NamearXiv.org [math.PR]
    Volume0802.3467

    Fingerprint Dive into the research topics of 'The Aizenman-Sims-Starr and Guerra's schemes for the SK model with multidimensional spins'. Together they form a unique fingerprint.

    Cite this