Universal microscopic correlation functions for products of independent Ginibre matrices

@article{Akemann2012UniversalMC,
  title={Universal microscopic correlation functions for products of independent Ginibre matrices},
  author={Gernot Akemann and Zdzislaw Burda},
  journal={arXiv: Mathematical Physics},
  year={2012}
}
We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all… 
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References

SHOWING 1-10 OF 25 REFERENCES
The chiral Gaussian two-matrix ensemble of real asymmetric matrices
We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought
Gap probabilities in non-Hermitian random matrix theory
We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles
Spectrum of the product of independent random Gaussian matrices.
  • Z. Burda, R. Janik, B. Waclaw
  • Mathematics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
TLDR
It is shown that the eigenvalue density of a product X=X1X2...XM of M independent NxN Gaussian random matrices in the limit N-->infinity is rotationally symmetric in the complex plane and it is conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill Pastur-Lindeberg's condition.
Non-Hermitian extensions of Wishart random matrix ensembles
We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart-Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian
Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices: The Extended Version
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit,
Induced Ginibre ensemble of random matrices and quantum operations
A generalization of the Ginibre ensemble of non-Hermitian random square matrices is introduced. The corresponding probability measure is induced by the ensemble of rectangular Gaussian matrices via a
Spectral relations between products and powers of isotropic random matrices.
We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of nth power of a single
Exact statistical properties of the zeros of complex random polynomials
The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of
Characteristic polynomials in real Ginibre ensembles
We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a
...
1
2
3
...