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We prove upper and lower bounds on the free energy in theSherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins interms of the variational inequalities based on the corresponding Parisifunctional. We employ the comparison scheme of Aizenman, Sims and Starr and theone of Guerra involving the generalised random energy model-inspired processesand Ruelle's probability cascades. For this purpose an abstract quenched largedeviations principle of the Gaertner-Ellis type is obtained. Using theproperties of Ruelle's probability cascades and the Bolthausen-Sznitmancoalescent, we derive Talagrand's representation of the Guerra remainder termfor our model. We study the properties of the multidimensional Parisifunctional by establishing a link with a certain class of the non-linearpartial differential equations. Solving a problem posed by Talagrand, we showthe strict convexity of the local Parisi functional. We prove the Parisiformula for the local free energy in the case of the multidimensional Gaussiana priori distribution of spins using Talagrand's methodology of the a prioriestimates.

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