Planck-scale number of nodal domains for toral eigenfunctions

TL;DR

This paper investigates the asymptotic behavior of the number of nodal domains in small balls for generic toral eigenfunctions, revealing that their count follows the same law as the global count at the Planck scale.

Contribution

It introduces a novel arithmetic approach to refine Bourgain's de-randomisation technique at Planck scale and proposes a Planck scale version of Yau's conjecture.

Findings

Nodal domain count obeys the same asymptotic law locally and globally.

Refinement of Bourgain's de-randomisation technique at Planck scale.

Proposal of a Planck scale version of Yau's conjecture.

Abstract

We study the number of nodal domains in balls shrinking slightly above the Planck scale for "generic" toral eigenfunctions. We prove that, up to the natural scaling, the nodal domains count obeys the same asymptotic law as the global number of nodal domains. The proof, on one hand, uses new arithmetic information to refine Bourgain's de-randomisation technique at Planck scale. And on the other hand, it requires a Planck scale version of Yau's conjecture which we believe to be of independent interest.