Diophantine Gaussian excursions and random walks

Abstract

We establish general asymptotic upper and lower bounds for the volume variance of Euclidean Gaussian nodal excursions in terms of the random walk associated to the spectral measure. These bound are sharp in several situations, and under mild assumptions, the variance is at least linear.

To obtain sublinear variances, we focus on the case where the spectral measure is purely atomic, and show that the associated irrational random walk on the multi-dimensional torus comes back more often close to 0 when the atoms are well approximable by rational tuples. Hence the excursion behaviour strongly depends on the diophantine properties of the atoms, i.e. on the quality of approximation of the atom locations by rationals. The volume variance has fluctuations which power can be arbitrarily close from the maximum $2d$ (quadratic fluctuations), whereas if the atoms are badly approximable the excursion is strongly hyperuniform, meaning the variance asymptotic power is minimal, $(d-1)$, corresponding to the window boundary measure. Also, given any reasonable variance asymptotic behaviour, there are uncountably many sets of spectral atoms that realise it.

The versatility of the variance formula is illustrated by other examples where the spectral measure support can have higher dimension, in particular it is able to capture the variance cancellation phenomenon of Gaussian random waves, and it also yields that there are no hyperuniform isotropic Gaussian excursions.