Critical points of multidimensional random Fourier series: variance estimates

Abstract

To any positive number $\varepsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function $u^\varepsilon$ on the $m$-torus $T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \sum_{\nu\in\mathbb{Z}^m} X_{\nu,\varepsilon} \exp\bigl(\; 2\pi \varepsilon \sqrt{-1} \;(\nu\cdot \theta)\;\bigr),$$ where $X_{\nu,\varepsilon}$ are complex independent Gaussian random variables with variance $w(\varepsilon|\nu|)$. Let $N^\varepsilon$ denote the number of critical points of $u^\varepsilon$. We describe explicitly two constants $C, C'$ such that as $\varepsilon$ goes to the zero, the expectation of the random variable $\frac{1}{{\rm vol}\,(T^m_\varepsilon)}N^\varepsilon$ converges to $C$, while its variance is extremely small and behaves like $C'\varepsilon^{m}$.