Complexity of Gaussian random fields with isotropic increments: critical points with given indices

TL;DR

This paper analyzes the asymptotic number of critical points with specific indices in a high-dimensional Gaussian random field with isotropic increments, relevant for understanding complex energy landscapes in statistical physics.

Contribution

It provides new asymptotic formulas for the mean number of critical points of given index in high-dimensional Gaussian fields with isotropic increments.

Findings

Derived formulas for the mean number of critical points with fixed index.

Extended analysis to include critical values in open sets.

Applicable to models of particles in random potentials.

Abstract

We study the landscape complexity of the Hamiltonian $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a smooth Gaussian process with isotropic increments on $\mathbb R^{N}$. This model describes a single particle on a random potential in statistical physics. We derive asymptotic formulas for the mean number of critical points of index $k$ with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis without the index constraint.