Critical sets of random smooth functions on compact manifolds

TL;DR

This paper studies the expected number of critical points of random smooth functions on compact manifolds, showing it scales linearly with the dimension of the function space and providing an explicit asymptotic constant.

Contribution

It establishes the asymptotic behavior of the expected critical points for random eigenfunction-based functions on manifolds, with explicit constants and growth rates.

Findings

Expected critical points scale as the dimension of the function space.

Explicit constant $C_m$ depends only on the manifold's dimension.

Asymptotic behavior of $C_m$ as dimension grows.

Abstract

Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\rightarrow \infty$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ and satisfying the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$ as $m\to \infty$.