The statistics of critical points of Gaussian fields on large-dimensional spaces

TL;DR

This paper computes the average number of critical points of Gaussian fields in high-dimensional spaces, revealing their organization and implications for glassy systems, disordered systems, and string theory landscapes.

Contribution

It provides a comprehensive calculation of critical point statistics in high-dimensional Gaussian fields, connecting mathematical results to physical theories.

Findings

Critical points distribution depends on energy and index.

Results inform understanding of energy landscapes in complex systems.

Applicable to theories in physics like string theory landscapes.

Abstract

We calculate the average number of critical points of a Gaussian field on a high-dimensional space as a function of their energy and their index. Our results give a complete picture of the organization of critical points and are of relevance to glassy and disordered systems, and to landscape scenarios coming from the anthropic approach to string theory.