Nodal Replication of Planar Random Waves

TL;DR

This paper investigates the near-periodic behavior of eigenmodes of flat planar manifolds at high energies, revealing that Gaussian Arithmetic Random Waves replicate almost identically at specific scales related to number theory, shedding light on nodal length correlations.

Contribution

It establishes precise scales at which eigenmodes replicate, connecting spectral properties with number-theoretic structures, and offers a heuristic for the minimal replication scale.

Findings

Eigenmodes replicate at scales involving sums of two squares

Correlation phenomena occur at larger scales than replication

Heuristic suggests minimal replication scale is closer to a smaller scale

Abstract

We study the almost periods of the eigenmodes of flat planar manifolds in the high energy limit. We prove in particular that the Gaussian Arithmetic Random Waves replicate almost identically at a scale at most ${\ell}$n := n -- 1 2 exp (Nn), where Nn is the number of ways n can be written as a sum of two squares. It provides a qualitative interpretation of the full correlation phenomenon of the nodal length, which is known to happen at scales larger than ${\ell}$ ' n := n --1/2 N A n. We provide also a heuristic with a toy model pleading that the minimal scale of replication should be closer to ${\ell}$ ' n than ${\ell}$n. Contents 1. Introduction 1 2. Almost periodicity and replication 6 3. Dirichlet's theorem for almost periodic fields 13 4. Replication of the nodal lines 15 5. Optimality of Dirichlet's approximation theorem 19 6. Appendix 20 References 28