The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins

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.

Parisi formula for the ground state energy in the mixed $p$-spin model

“…To overcome this issue, we construct a representation of the Parisi PDE in terms of the stochastic optimal control problem introduced in [2,3]. Under this framework, we are able to deal with the large β limit of the Parisi functional and we remove this singularity as a marvel cancellation happens between the non-linear PDE and the linear term in the functional P β .…”

“…See a simplified argument of [2] in [8]. Different than the derivations in [2,3,8], here we present an approach that relies only on Itô's formula. Consider the following Parisi PDE, It is known [1,8] that given (A1), the solution Ψ has the properties that ∂ j x Ψ ∈ C([0, 1] × R) for all j ≥ 0 and |∂ x Ψ| ≤ 1.…”

“…Remark 3. In [2,3,8], the fact that the process u * attains the maximum value of F s,t (·, x) was established by a direct verification using Itô's formula. From the above proof, the equation (8) quantifies the distance between the PDE and the functional F s,t for any u.…”

“…Variational representation for the Parisi PDEIn this section, we will derive a variational representation for the PDE (1) in the form of stochastic optimal control. This formulation appeared initially in[3] and was used to establish…”

High-dimensional Gaussian fields with isotropic increments seen through spin glasses

“…In the high-dimensional limit, we derive a computable saddle-point representation for the free energy, which is similar to the Parisi formula for the Sherrington-Kirkpatrick (SK) model of a mean-field spin glass. Our proofs are based on the local comparison arguments for Gaussian fields with non-constant variance developed in [5], which are, in turn, based on the ideas of Guerra [9], Guerra and Toninelli [10], Talagrand [16] and Panchenko [13].…”