Electronic Journal of Probability

Powers of Ginibre eigenvalues

Guillaume Dubach

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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

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

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

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Zentralblatt MATH identifier

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

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

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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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