Permanental processes from products of complex and quaternionic induced Ginibre ensembles
We consider products of independent random matrices taken from the inducedGinibre ensemble with complex or quaternion elements. The joint densities forthe complex eigenvalues of the product matrix can be written down exactly for aproduct of any fixed number of matrices and any finite matrix size. We showthat the squared absolute values of the eigenvalues form a permanental process,generalising the results of Kostlan and Rider for single matrices to productsof complex and quaternionic matrices. Based on these findings, we can firstwrite down exact results and asymptotic expansions for the so-called holeprobabilities, that a disc centered at the origin is void of eigenvalues.Second, we compute the asymptotic expansion for the opposite problem, that alarge fraction of complex eigenvalues occupies a disc of fixed radius centeredat the origin; this is known as the overcrowding problem. While the expressionsfor finite matrix size depend on the parameters of the induced ensembles, theasymptotic results agree to leading order with previous results for products ofsquare Ginibre matrices.
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