Critical points of multidimensional random Fourier series: central limits

Abstract

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 $\hbar\searrow 0$.