Non-Universal Moderate Deviation Principle for the Nodal Length of Arithmetic Random Waves

TL;DR

This paper establishes a non-universal moderate deviation principle for the nodal length of arithmetic random waves on flat tori, revealing new probabilistic behaviors in the fluctuations of these eigenfunctions.

Contribution

It introduces a simple proof of a non-universal moderate deviation principle for the nodal length, connecting long memory effects and chaotic expansions in arithmetic random waves.

Findings

Proves a non-central moderate deviation principle for nodal length.

Shows non-universality of deviation behavior in arithmetic random waves.

Utilizes techniques from large deviation theory and chaotic analysis.

Abstract

Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established in [MPRW16] and [BMW20] respectively, by means of chaotic expansions, number theoretical estimates and full correlation phenomena. Our proof is simple and relies on the interplay between the long memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well as on well-known techniques in Large Deviation theory (the contraction principle and the concept of exponential equivalence).