On the Euler-Poincar\'e Characteristic of the planar Berry's random wave: fluctuations and a perturbation study

TL;DR

This paper establishes the Gaussian fluctuation behavior of the Euler-Poincaré characteristic in Berry's random wave models, including perturbed versions, supported by theoretical proofs, explicit variance calculations, and numerical analysis.

Contribution

It proves the CLT for the Euler-Poincaré characteristic in Berry's random wave models and explores Gaussian fluctuations in perturbed models with detailed statistical and numerical analysis.

Findings

Proved CLT for Euler-Poincaré characteristic in Berry's wave models

Demonstrated Gaussian fluctuations in perturbed Berry's models

Provided explicit variance calculations and numerical support

Abstract

We prove the Central Limit Theorem for the Euler-Poincar\'e characteristic of Berry's random wave model in a growing domain. We also show Gaussian fluctuations for a class of Berry's mixture models that correspond to a perturbation of the initial random field. Finally, some statistical applications, explicit calculations of the variance of the perturbed Berry's model and numerical investigations are provided to support our theoretical results.