Hamilton-Jacobi equations from mean-field spin glasses

Abstract

We establish the well-posedness of Hamilton-Jacobi equations arising from meanfield spin glass models in the viscosity sense. Originally defined on the set of monotone probability measures, these equation can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf-Lax formula, or the Hopf formula on cones. As the first step, we show the well-posedness of equations on finite-dimensional cones, which is self-contained and, we believe, is of independent interest. The surprising observation key to our work is that a monotonicity assumption on the nonlinearity allows us to prescribe that the solution satisfies the equation in the viscosity sense at any point, without having to distinguish between interior and boundary points.Previously, two notions of solutions were considered, one defined directly as the Hopf-Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

“…In this aspect of our proof, we deviate from available results in [13]. In contrast, [13] establishes the equality of the right sides via the Hopf-Lax representation of the solution of an underlying Hamilton-Jacobi equation (see also [14,20,36]).…”

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“…It should be noted that the Hamilton-Jacobi equation mentioned here is a finitedimensional one, rather than the infinite-dimensional one in [15,16,17,5] associated with the enriched model discussed in Section 6. The former is similar to the one related to the Curie-Weiss model described in [14].…”