Fluctuations of Level Curves for Time-Dependent Spherical Random Fields

TL;DR

This paper studies the fluctuations of level curves in time-dependent isotropic Gaussian spherical random fields, revealing different behaviors under long and short memory assumptions, including dominance of a single chaos component and a CLT for all chaoses.

Contribution

It extends the analysis of geometric functional fluctuations to time-dependent spherical fields, identifying dominant components and establishing CLTs under different memory conditions.

Findings

In long memory, fluctuations are dominated by a single chaos component.

Cancellation points exist where variance is smaller, excluding the nodal case.

In short memory, all chaoses contribute, and a CLT is proved.

Abstract

The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of $u$-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case $u = 0$, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer-Major argument.