A lower bound for the Bogomolny-Schmit constant for random monochromatic plane waves

TL;DR

This paper establishes a lower bound for the Bogomolny-Schmit constant related to the expected number of nodal domains in large disks for random monochromatic plane waves, advancing understanding of their asymptotic behavior.

Contribution

It provides an elementary method to derive a lower bound for the Bogomolny-Schmit constant, a key proportionality factor in the expected count of nodal domains.

Findings

Derived a lower bound for the Bogomolny-Schmit constant

Enhanced mathematical understanding of nodal domain distribution

Utilized elementary techniques for the bound estimation

Abstract

This note deals with nodal domains of random monochromatic plane waves. It was shown by Nazarov and Sodin that the expected number of such nodal domains included in a disk of radius $R$ is proportional to $\pi R^2$ in the large $R$ limit. However, very little is known on the value of the proportionality constant from a mathematical point of view. The aim of this note is to obtain a lower bound on the value of this constant my elementary means.