A law of large numbers concerning the number of critical points of isotropic Gaussian functions

TL;DR

This paper proves a law of large numbers for the distribution of critical points of isotropic Gaussian functions, showing convergence to a deterministic measure as the domain size grows infinitely large.

Contribution

It establishes a rigorous asymptotic law for the distribution of critical points of isotropic Gaussian functions, including convergence in probability and in L^2.

Findings

Critical point distributions converge to Lebesgue measure as domain size increases

Provided precise asymptotics for second moments of critical point counts

Demonstrated almost sure and L^2 convergence of the distributions

Abstract

We investigate the distribution of critical points of certain isotropic random functions $\Phi$ on $\mathbb{R}^m$. We show that the distribution of critical points of $\Phi(Rx)$, suitably normalized, converge a.s. and $L^2$ as random measures to the (deterministic) Lebesgue measure as $R\to\infty$. We achieve this by producing precise asymptotics of the second moments of these distributions as $R\to\infty$.