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

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.