Critical points of multidimensional random Fourier series: central limits

TL;DR

This paper studies the critical points of Gaussian random Fourier series on high-dimensional tori, showing their count follows a predictable pattern with a central limit theorem as the frequency parameter tends to infinity.

Contribution

It provides the first rigorous analysis of the asymptotic distribution of critical points for multidimensional random Fourier series, including mean, variance, and CLT results.

Findings

Mean number of critical points scales as rac{1}{\u03b7^m}

Variance scales as rac{1}{rac{m}{2}}

Critical points distribution converges to a normal distribution

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 $\hbar\searrow 0$.