Electronic Journal of Probability

Powers of Ginibre eigenvalues

Guillaume Dubach

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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.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 111, 31 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1540865374

Digital Object Identifier
doi:10.1214/18-EJP234

Zentralblatt MATH identifier
06970416

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices

Keywords
complex Ginibre ensemble conditional independence power maps radially symmetric determinantal point process Gaussian Free Field beta ensembles

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dubach, Guillaume. Powers of Ginibre eigenvalues. Electron. J. Probab. 23 (2018), paper no. 111, 31 pp. doi:10.1214/18-EJP234. https://projecteuclid.org/euclid.ejp/1540865374


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