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We study the free energy of a particle in (arbitrary) high-dimensionalGaussian random potentials with isotropic increments. We prove a computablesaddle-point variational representation in terms of a Parisi-type functionalfor the free energy in the infinite-dimensional limit. The proofs are based onthe techniques developed in the course of the rigorous analysis of theSherrington-Kirkpatrick model with vector spins.

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