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

TL;DR

This paper investigates the free energy of particles in high-dimensional isotropic Gaussian potentials, deriving a variational formula using spin glass techniques, advancing understanding of complex random systems.

Contribution

It introduces a Parisi-type variational representation for the free energy of high-dimensional Gaussian fields with isotropic increments, extending spin glass methods.

Findings

Derived a saddle-point variational formula for the free energy.

Connected the analysis to the rigorous study of the Sherrington-Kirkpatrick model.

Provided a computable approach for high-dimensional Gaussian potentials.

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.