Moderate Deviation estimates for Nodal Lengths of Random Spherical Harmonics

TL;DR

This paper establishes moderate deviation estimates for the nodal lengths of random spherical harmonics on the entire sphere and shrinking domains, extending previous CLT results with a new probabilistic approach.

Contribution

It introduces a novel application of Moderate Deviation Principles to the study of nodal lengths of random spherical harmonics, combining Wiener chaos analysis with exponential equivalence.

Findings

Moderate deviation estimates are proven for nodal lengths on the sphere.

Results extend CLTs to moderate deviation regimes.

Method combines Wiener chaos techniques with exponential equivalence.

Abstract

We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Th\"ale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.