Abstract
We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle point variational representation in terms of a Parisi-type functional for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.
Highlights
We study the free energy of a particle in high-dimensional Gaussian random potentials with isotropic increments
Considerable attention in the theoretical physics literature has been devoted to Gaussian random fields with isotropic increments viewed as random potentials, see, e.g, the works by Fyodorov and Sommers [8], Fyodorov and Bouchaud [7], and references therein
We concentrate on the computation of the free energy of a particle subjected to arbitrary high-dimensional Gaussian random potentials with isotropic increments
Summary
Considerable (renewed) attention in the theoretical physics literature has been devoted to Gaussian random fields with isotropic increments viewed as random potentials, see, e.g, the works by Fyodorov and Sommers [8], Fyodorov and Bouchaud [7], and references therein. It was heuristically argued in these works that Parisi’s theory of hierarchical replica symmetry breaking (Parisi Ansatz, cf [11]) is applicable in this context. In the Appendix, we provide some complementary information for the reader’s convenience
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