Diophantine Gaussian excursions and random walks

TL;DR

This paper studies the asymptotic behavior of Gaussian nodal excursions and their variance in relation to the diophantine properties of spectral measure atoms, revealing how approximation quality influences variance fluctuations.

Contribution

It establishes a link between diophantine approximation properties of spectral measure atoms and the variance asymptotics of Gaussian nodal excursions, including bounds and variability conditions.

Findings

Variance fluctuations depend on diophantine properties of spectral atoms.

Badly approximable ratios lead to bounded variance by boundary measure.

Uncountably many parameter sets yield prescribed variance asymptotics.

Abstract

We investigate the asymptotic variance of Gaussian nodal excursions in the Euclidean space, focusing on the case where the spectral measure has incommensurable atoms. This study requires to establish fine recurrence properties in 0 for the associated irrational random walk on the torus. We show in particular that the recurrence magnitude depends strongly on the diophantine properties of the atoms, and the same goes for the variance asymptotics of nodal excursions. More specifically, if the spectral measures contains atoms which ratios are well approximable by rationals, the variance is likely to have large fluctuations as the observation window grows, whereas the variance is bounded by the (d -- 1)-dimensional measure of the window boundary if these ratio are badly approximable. We also show that, given any reasonable function, there are uncountably many sets of parameters for which the variance is asymptotically equivalent to this function.