Critical points of multidimensional random Fourier series: Central limits

Abstract

We investigate certain families $X^{\hbar}$, $0<\hbar\ll1$ of stationary Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^{m}:=\mathbb{R}^{m}/\mathbb{Z}^{m}$ approaching the white noise as $\hbar\to0$. We show that there exists universal constants $c_{1},c_{2}>0$ such that for any cube $B\subset\mathbb{R}^{m}$ of size $r\leq1/2$, the number of critical points of $X^{\hbar}$ in the region $B\bmod\mathbb{Z}^{m}\subset\mathbb{T}^{m}$ has mean $\sim c_{1}\operatorname{vol}(B)\hbar^{-m}$, variance $\sim c_{2}\operatorname{vol}(B)\hbar^{-m}$, and satisfies a central limit theorem as $\hbar\searrow0$.