A law of large numbers concerning the distribution of critical points of random Fourier series
Qiangang "Brandon'' Fu, Liviu I. Nicolaescu
TL;DR
The authors analyze the distribution of critical points of random Fourier series on the flat torus by shaping a white-noise limit via $F^R_\mathfrak{a}=\mathfrak{a}(R^{-1}\sqrt{\Delta})W$. They prove that for test functions supported in geodesic balls, the variance of the critical-point count scales as $R^{m}$ and that, for $m\ge 2$, the scaled random measures converge almost surely to a deterministic multiple of the volume measure. The methodology blends Kac-Rice calculus for Gaussian fields with precise covariance-approximation estimates and a radial blow-up analysis to separate diagonal from off-diagonal contributions. The results provide a functional strong law of large numbers for the random critical-point measures and connect finite-$R$ statistics to the isotropic white-noise limit, highlighting universal variance constants $V_m(\mathfrak{a})$ and intensity constants $C_m(\mathfrak{a})$. This advances understanding of high-dimensional random fields by quantifying equidistribution of critical points in the white-noise regime.
Abstract
On the flat torus $\mathbb{T}^m=\mathbb{R}^m/\mathbb{Z}^m$ with angular coordinates $\vecθ$ we consider the random function $F_R=\mathfrak{a}\big(\, R^{-1} \sqrtΔ\,\big) W$, where $R>0$, $Δ$ is the Laplacian on this flat torus, $\mathfrak{a}$ is an even Schwartz function on $\mathbb{R}$ such that $\mathfrak{a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{T}^m)$ we set \[ Z_R(f):=\sum_{\nabla F_R(\vecθ)=0} f(\vecθ) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{-m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus.