Powers of Ginibre Eigenvalues

TL;DR

This paper investigates how complex Ginibre eigenvalues behave under power maps, revealing a decomposition into independent determinantal point processes and establishing connections with other random matrix ensembles and Gaussian fields.

Contribution

It introduces the Power-Ginibre distributions, generalizes Rains' superposition theorem, and extends fluctuation results to new classes of radially symmetric ensembles.

Findings

Eigenvalues under power maps decompose into independent processes.

Fluctuations of linear statistics converge to Gaussian variables.

Results extend to radially symmetric normal matrix ensembles.

Abstract

We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. The decomposition can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE and Kostlan's independence of radii to a wider class of point processes. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE. Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of their smooth linear statistics to independent gaussian variables, coherent with the link between the complex Ginibre Ensemble and the Gaussian Free Field. Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $\beta$ are discussed, replacing independence by conditional independence.