On the number of nodal domains of toral eigenfunctions

TL;DR

This paper investigates the asymptotic behavior of the number of nodal domains in eigenfunctions on a torus, establishing optimal lower bounds for generic cases using probabilistic and de-randomisation techniques.

Contribution

It provides the first precise asymptotic results for nodal domains of generic toral eigenfunctions, extending Nazarov-Sodin's framework to this setting.

Findings

Establishes a precise asymptotic for the number of nodal domains in generic eigenfunctions.

Provides an optimal lower bound for the count of nodal domains.

Extends probabilistic methods to deterministic eigenfunctions on the torus.

Abstract

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.