Complexity of Gaussian random fields with isotropic increments

TL;DR

This paper analyzes the topology of level sets of high-dimensional Gaussian random fields with isotropic increments, deriving formulas for the expected number of critical points as dimension grows.

Contribution

It provides asymptotic formulas for the mean number of critical points in high dimensions for Gaussian fields with isotropic increments, extending understanding of their energy landscapes.

Findings

Asymptotic formulas for the mean number of critical points

Analysis of the energy landscape topology in high dimensions

Extension to critical points with specified index in a companion paper

Abstract

We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on $\mathbb R^{N}$ of the form $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a Gaussian process with isotropic increments. We derive asymptotic formulas for the mean number of critical points with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis for the number of critical points with a given index.