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2018 Powers of Ginibre eigenvalues
Guillaume Dubach
Electron. J. Probab. 23: 1-31 (2018). DOI: 10.1214/18-EJP234

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.

Citation

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Guillaume Dubach. "Powers of Ginibre eigenvalues." Electron. J. Probab. 23 1 - 31, 2018. https://doi.org/10.1214/18-EJP234

Information

Received: 5 January 2018; Accepted: 9 October 2018; Published: 2018
First available in Project Euclid: 30 October 2018

zbMATH: 06970416
MathSciNet: MR3878136
Digital Object Identifier: 10.1214/18-EJP234

Subjects:
Primary: 15B52, 60B20

JOURNAL ARTICLE
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