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

On the flat torus ${\mathbb{T}}^m={\mathbb{R}}^m\mathrm{/}{\mathbb{Z}}^m$ with angular coordinates $\vec{\theta }$ we consider the random function $F_R=\mathfrak{a}\left(R^{-1}\sqrt{\mathrm{\Delta }}\right)W$, where $R>0$, $\mathrm{\Delta }$ 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 $$Z_R(f):=\sum _{\mathrm{\nabla }{F}_R(\vec{\theta })=0}f(\vec{\theta })$$ 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 $R\to \mathrm{\infty }$. We use this to prove that if $m\ge 2$, then as $N\to \mathrm{\infty }$ the random measures $N^{-m}{Z}_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus.