  To any positive number $\varepsilon$ and any nonnegative even Schwartz
function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function
$u^\varepsilon$ on the $m$-torus
$T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the
real part of the random Fourier series $$ \sum_{\nu\in\mathbb{Z}^m}
X_{\nu,\varepsilon} \exp\bigl(\; 2\pi \varepsilon \sqrt{-1} \;(\nu\cdot
\theta)\;\bigr),$$ where $X_{\nu,\varepsilon}$ are complex independent Gaussian
random variables with variance $w(\varepsilon|\nu|)$. Let $N^\varepsilon$
denote the number of critical points of $u^\varepsilon$. We describe explicitly
two constants $C, C'$ such that as $\varepsilon$ goes to the zero, the
expectation of the random variable $\frac{1}{{\rm
vol}\,(T^m_\varepsilon)}N^\varepsilon$ converges to $C$, while its variance is
extremely small and behaves like $C'\varepsilon^{m}$.
