Non-universality of the Nazarov-Sodin constant

TL;DR

This paper demonstrates that the Nazarov-Sodin constant varies with the specific Gaussian field, impacting the expected count of nodal components, and extends this result to arithmetic random waves on the torus.

Contribution

It proves the non-universality of the Nazarov-Sodin constant and applies this to arithmetic random waves, showing dependence on the underlying field.

Findings

Nazarov-Sodin constant depends on the Gaussian field

The result applies to arithmetic random waves on the torus

Expected number of nodal components varies with the field

Abstract

We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.