Powers of Ginibre eigenvalues

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, \[ \mathrm{Gin} (N)^M \stackrel{d} {=} \bigcup _{k=1}^M \mathrm{Gin} (N,M,k), \] where the so-called Power-Ginibre distributions $\mathrm{Gin} (N,M,k)$ 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 (see [21]) and Kostlan’s independence of radii (see [17]) to a wider class of point processes. Our proof technique also allows us to recover two results by Edelman and La Croix [12] 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 [22].

Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $\beta $ are discussed, replacing independence by conditional independence.