Journal article

Permanental processes from products of complex and quaternionic induced Ginibre ensembles

Gernot Akemann, Jesper R Ipsen, Eugene Strahov

Random Matrices: Theory and Applications | WORLD SCI PUBL CO INC | Published : 2014

Abstract

We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalizing the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disk centered at the origin is void of eigenvalues. Second, w..

View full abstract

University of Melbourne Researchers

Grants

Awarded by "Symmetries and Universality in Mesoscopic Systems" of the German research council DFG & DAAD International Network "From Extreme Matter to Financial Markets"


Awarded by "Stochastics and Real World Models"


Funding Acknowledgements

We would like to thank Alon Nishry for discussions. The SFB|TR12 "Symmetries and Universality in Mesoscopic Systems" of the German research council DFG & DAAD International Network "From Extreme Matter to Financial Markets" (G. Akemann), "Stochastics and Real World Models" IRTG 1132 (J. R. Ipsen) are acknowledged for financial support. The School of Mathematics at the IAS Princeton is thanked for its kind hospitality where part of this work was written up.