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.
Highlights
The complex Ginibre ensemble, that we will denote by Gin(N ), consists of matrices whose coefficients are independent and identically distributed complex Gaussian random variables
Instead of relying on the underlying matrix ensemble, or decomposing the density, we rely on a characterization of the law by the statistics obtained with the so-called product symmetric polynomials, that is, expressions of the type i=1 for any polynomial P in two variables. Such statistics can be exactly computed thanks to Andréief’s identity, and they characterize the point process. We illustrate this in Subsection 2.1 by giving a new proof of Kostlan’s theorem and the independence of M th powers for M ≥ N
For the convenience of the reader, we provide a table summing up all results considered in our paper about three ensembles of random matrix theory: namely, the complex Ginibre ensemble, the Circular Unitary Ensemble, and the Gaussian Unitary Ensemble
Summary
The complex Ginibre ensemble, that we will denote by Gin(N ), consists of matrices whose coefficients are independent and identically distributed complex Gaussian random variables. Two other results hinted that something unusual happens with quadratic repulsion that would concern more than the radii or the high powers only. The first of these results was stated for the eigenvalues of a Haar-distributed unitary matrix (known as the Circular Unitary Ensemble, or CUE). GUE eigenvalues are distributed on R with joint density proportional to:. GUE(N )2 =d LUE(N, 1) ∪ LUE(N, 2), where LUE(N, k) stand for Laguerre ensembles with half-integer parameters The fact that such a result holds for the squares only is essentially due to the lack of symmetry
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