A Central Limit Theorem for Fluctuations in Polyanalytic Ginibre Ensembles

TL;DR

This paper proves a central limit theorem for fluctuations in Polyanalytic Ginibre ensembles, revealing asymptotic normality and variance stabilization across Landau levels, extending previous results for analytic cases.

Contribution

It introduces a CLT for fluctuations in polyanalytic Ginibre ensembles and analyzes variance contributions from bulk and boundary, extending known results to higher Landau levels.

Findings

Fluctuations are asymptotically normal.

Variance has independent bulk and boundary components.

Combining Landau levels stabilizes the variance.

Abstract

We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Vir\'ag. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure $q$-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure $q$-analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.