Critical points of multidimensional random Fourier series: central limits

Abstract

We investigate certain families $X^{\hslash }$, $0<\hslash \ll 1$, of Gaussian random smooth functions on the $m$-dimensional torus ${\mathbb{T}}_{\hslash }^m:={\mathbb{R}}^m/({\hslash }^{-1}\mathbb{Z}{)}^m$. 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 $X^{\hslash }$ in the region ${\hslash }^{-1}B/({\hslash }^{-1}\mathbb{Z}{)}^m\subset {\mathbb{T}}_{\hslash }^m$ has mean $\sim c_1{\hslash }^{-m}$, variance $\sim c_2{\hslash }^{-m/2}$, $c_1,c_2>0$, and satisfies a central limit theorem as $\hslash \searrow 0$.

34 pages,1 figure added a new result, improved the presentation

References in corpus (3)

• Fluctuations of the nodal length of random spherical harmonics, erratum
• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
• On the distribution of the nodal sets of random spherical harmonics