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

We prove the Central Limit Theorem for the Euler-Poincaré 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.