Complexity of random smooth functions on compact manifolds

TL;DR

This paper explores the relationship between eigenvalue distributions of random matrices and critical values of random linear combinations of Laplacian eigenfunctions on compact manifolds, establishing a central limit theorem for high-dimensional cases.

Contribution

It introduces a novel connection between random matrix eigenvalues and Laplacian eigenfunction critical values, and proves a central limit theorem for large-dimensional manifolds.

Findings

Eigenvalue distribution of GOE matrices relates to Laplacian eigenfunction critical values.

Established a central limit theorem for high-dimensional manifolds.

Provides insights into the complexity of random smooth functions on manifolds.

Abstract

We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimension of the manifold is very large.