Complexity of Gaussian random fields with isotropic increments

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 $X_N(x)+\frac{\mu }{2}\Vert x{\Vert }^2,$ 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.

arXiv commentOriginal version was split into two papers. This is part one. Minor updates. 35 pages