On statistical mechanics of a single particle in high-dimensional random landscapes

TL;DR

This paper explores the statistical mechanics of a single particle in high-dimensional Gaussian landscapes, focusing on landscapes with logarithmic correlations and their multifractal structures, along with related stationary point counting and local landscape models.

Contribution

It provides new insights into the multifractal structure of Boltzmann-Gibbs measures in high-dimensional random landscapes with logarithmic correlations.

Findings

Multifractal spatial structure of the Boltzmann-Gibbs measure.

Analysis of landscapes with logarithmic correlations.

Results on counting stationary points of Gaussian surfaces.

Abstract

We discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(>>1)-dimensional Gaussian landscape. The particular attention is paid to the case of landscapes with logarithmically growing correlations and to its recent generalisations. Those landscapes give rise to a rich multifractal spatial structure of the associated Boltzmann-Gibbs measure. We also briefly mention related results on counting stationary points of random Gaussian surfaces, as well as ongoing research on statistical mechanics in a random landscape constructed locally by adding many squared Gaussian-distributed terms.