# Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann G, Cikovic M (2017)
arXiv:1711.05488.

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We study the complex eigenvalues of the product of a fixed number of independent, complex non-Hermitian random matrices at finite matrix size. The factors are taken from the induced Ginibre ensemble with or without a fixed trace constraint, representing a microcanonical, non-Gaussian distribution. We start with a single fixed trace ensemble, comparing the constraint for induced Ginibre and Gaussian normal random matrices. Despite finding different non-determinantal joint densities for these two ensembles, all $k$-point correlation functions resulting from an inverse Laplace transform of the known ensembles without constraint agree, up to a simple rescaling. Because products of normal random matrices are difficult even without posing a constraint, we then focus on the product of $m$ induced Ginibre matrices with fixed trace constraint and $M-m$ without. Using an $m$-fold inverse Laplace transform, we obtain a concise result for the spectral density of such products at finite matrix size, for arbitrary fixed $m$ and $M$. Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble. Here, we argue that the density for mixed products is in the same universality class as the product of $M$ independent induced Ginibre ensembles.
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Akemann G, Cikovic M. Products of random matrices from fixed trace and induced Ginibre ensembles. arXiv:1711.05488. 2017.
Akemann, G., & Cikovic, M. (2017). Products of random matrices from fixed trace and induced Ginibre ensembles. arXiv:1711.05488
Akemann, G., and Cikovic, M. (2017). Products of random matrices from fixed trace and induced Ginibre ensembles. arXiv:1711.05488.
Akemann, G., & Cikovic, M., 2017. Products of random matrices from fixed trace and induced Ginibre ensembles. arXiv:1711.05488.
G. Akemann and M. Cikovic, “Products of random matrices from fixed trace and induced Ginibre ensembles”, arXiv:1711.05488, 2017.
Akemann, G., Cikovic, M.: Products of random matrices from fixed trace and induced Ginibre ensembles. arXiv:1711.05488. (2017).
Akemann, Gernot, and Cikovic, Milan. “Products of random matrices from fixed trace and induced Ginibre ensembles”. arXiv:1711.05488 (2017).
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