A law of large numbers concerning the distribution of critical points of random Fourier series

TL;DR

This paper investigates the distribution of critical points of certain random Fourier series on flat tori, establishing a law of large numbers for their critical point measures as the frequency parameter grows.

Contribution

It proves a law of large numbers for the distribution of critical points of random Fourier series on flat tori, including variance asymptotics and almost sure convergence results.

Findings

Variance of critical point sums grows like R^m

Critical point measures converge to volume measure almost surely

Explicit asymptotic formulas for variance and limits

Abstract

On the flat torus $\mathbb{T}^m=\mathbb{R}^m/\mathbb{Z}^m$ with angular coordinates $\vec{\theta}$ we consider the random function $F_R=\mathfrak{a}\big(\, R^{-1} \sqrt{\Delta}\,\big) W$, where $R>0$, $\Delta$ is the Laplacian on this flat torus, $\mathfrak{a}$ is an even Schwartz function on $\mathbb{R}$ such that $\mathfrak{a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{T}^m)$ we set \[ Z_R(f):=\sum_{\nabla F_R(\vec{\theta})=0} f(\vec{\theta}) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{-m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus.