A note on $3d$-monochromatic random waves and cancellation

TL;DR

This paper proves that the variance of the nodal length of 3D monochromatic random waves grows linearly with volume, establishing a CLT, and compares it with 2D cases where faster divergence occurs due to Berry's cancellation.

Contribution

It demonstrates linear variance growth and a CLT for 3D monochromatic random waves, extending understanding of their statistical behavior.

Findings

Variance of nodal length is linear in volume in 3D.

A CLT holds for 3D monochromatic random waves.

Comparison shows faster divergence in 2D due to Berry's cancellation.

Abstract

In this note we prove that the asymptotic variance of the nodal length of complex-valued monochromatic random waves restricted to an increasing domain in $\R^3$ is linear in the volume of the domain. Put together with previous results this shows that a Central Limit Theorem holds true for $3$-dimensional monochromatic random waves. We compare with the variance of the nodal length of the real-valued $2$-dimensional monochromatic random waves where a faster divergence rate is observed, this fact is connected with Berry's cancellation phenomenon. Moreover, we show that a concentration phenomenon takes place.