On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces

TL;DR

This paper constructs a Gaussian landscape with multiscale correlations in finite dimensions and shows its free energy behavior aligns with Derrida's GREM in high dimensions, with implications for understanding complex energy landscapes.

Contribution

It provides an explicit construction of Parisi landscapes in finite-dimensional Euclidean spaces and links their thermodynamic properties to the GREM model.

Findings

Free energy matches GREM in high dimensions

Low-temperature behavior depends on landscape's scale spectrum

Construction valid from 1D to high dimensions

Abstract

We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N>>1 the free energy of the system in the thermodynamic limit coincides with the most general version of Derrida's Generalized Random Energy Model. The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. We argue that our construction is in fact valid in any finite spatial dimensions, starting from N=1.