Critical points of multidimensional random Fourier series: variance estimates

Abstract

To any positive number $\epsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to \mathbb{R}$ we associate the random function $u^{\epsilon }$ on the $m$-torus $T_{\epsilon }^m:={\mathbb{R}}^m/({\epsilon }^{-1}\mathbb{Z}{)}^m$ defined as the real part of the random Fourier series $$\sum _{\nu \in {\mathbb{Z}}^m}{X}_{\nu ,\epsilon }\exp (2\pi \epsilon \sqrt{-1}(\nu \cdot \theta )),$$ where $X_{\nu ,\epsilon }$ are complex independent Gaussian random variables with variance $w(\epsilon |\nu |)$. Let $N^{\epsilon }$ denote the number of critical points of $u^{\epsilon }$. We describe explicitly two constants $C,C^{\prime }$ such that as $\epsilon$ goes to the zero, the expectation of the random variable $\frac{1}{\mathrm{vol}(T_{\epsilon }^m)}{N}^{\epsilon }$ converges to $C$, while its variance is extremely small and behaves like $C^{\prime }{\epsilon }^m$.

44 pages. Fixed typos, improved presentation, added references

References in corpus (1)

• Random Morse functions and spectral geometry

Cited by in corpus (2)

• High-Dimensional Random Fields and Random Matrix Theory
• Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials