# Weak commutation relations and eigenvalue statistics for products of rectangular random matrices

Ipsen J, Kieburg M (2014)

Physical Review E 89(3).

*Journal Article*|

*Published*|

*English*

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Abstract

We study the joint probability density of the eigenvalues of a product of rectangular real, complex, or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary, or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously has been discussed for infinite matrix size. Moreover, we derive the joint probability densities of the eigenvalues. To illustrate our results, we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights, we show that the product of complex random matrices yields a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.

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Ipsen J, Kieburg M. Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

*Physical Review E*. 2014;89(3).Ipsen, J., & Kieburg, M. (2014). Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

*Physical Review E*,*89*(3).Ipsen, J., and Kieburg, M. (2014). Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

*Physical Review E*89.Ipsen, J., & Kieburg, M., 2014. Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

*Physical Review E*, 89(3).J. Ipsen and M. Kieburg, “Weak commutation relations and eigenvalue statistics for products of rectangular random matrices”,

*Physical Review E*, vol. 89, 2014.Ipsen, J., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Physical Review E. 89, (2014).

Ipsen, Jesper, and Kieburg, Mario. “Weak commutation relations and eigenvalue statistics for products of rectangular random matrices”.

*Physical Review E*89.3 (2014).
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PMID: 24730789

PubMed | Europe PMC

arXiv 1310.4154

Inspire 1260609