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 $F_R=\mathfrak{a}\left(R^{-1}\sqrt{\Delta }\right)W$, 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 \[ Z_R(f):=\sum_{\nabla F_R(\vecθ)=0} f(\vecθ) \] 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 $m\ge 2$, 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.

32 pages