The Aizenman-Sims-Starr and Guerra’s schemes for the SK model with multidimensional spins. (English) Zbl 1205.60166

The authors deal with the question of the validity of the Parisi formula in the case where spins take values in a \(d\)-dimensional Riemannian manifold. Upper and lower bounds on the free energy of the Sherrington-Kirkpatrick model with multidimensional spins in terms of variational inequalities are given. The bounds are based on a multidimensional extension of the Parisi functional. The comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle’s probability cascades is generalized and unified. To do this, an abstract quenched large deviations principle of the Gartner-Ellis type is obtained. Talagrand’s representation of Guerra’s remainder term for the Sherrington-Kirkpatrick model with multidimensional spins is derived. The derivation is based on well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. Several properties of the multidimensional Parisi functional are studied by establishing a link with a certain class of semi-linear partial differential equations. The problem of strict convexity of the Parisi functional is transferred to a more general setting and the convexity in some particular cases is proved which sheds some light on the original convexity problem of Talagrand. Finally, the Parisi formula for the local free energy is established in the case of multidimensional Gaussian, a priori distribution of spins using Talagrand’s methodology of a priori estimates.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60F10 Large deviations
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