  We investigate certain families $X^\hbar$, $0<\hbar \ll 1$, of Gaussian
random smooth functions on the $m$-dimensional torus
$\mathbb{T}^m_\hbar:=\mathbb{R}^m/(\hbar^{-1}\mathbb{Z} )^m$. We show tha,t for
any cube $B\subset \mathbb{R}^m$ of size $<1/2$ and centered at the origin, the
number of critical points of $X^\hbar$ in the region
$\hbar^{-1}B/(\hbar^{-1}\mathbb{Z} )^m\subset\mathbb{T}^m_\hbar$ has mean $\sim
c_1\hbar^{-m}$, variance $\sim c_2\hbar^{-m/2}$, $c_1,c_2>0$, and satisfies a
central limit theorem as $\hbar\searrow 0$.
