Permanental processes from products of complex and quaternionic induced Ginibre ensembles

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, generalising 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 disc centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disc of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.

arXiv comment47 pages, v2: typos corrected, 1 reference added, published version

References in corpus (15)

• Determinantal Processes and Independence
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Spectrum of the Product of Independent Random Gaussian Matrices
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• The Cauchy two-matrix model
• Eigenvalue correlations in non-Hermitean symplectic random matrices
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• On the Asymptotic Spectrum of Products of Independent Random Matrices
• Products of independent elliptic random matrices
• Differential equations for singular values of products of Ginibre random matrices
• Products of Independent Quaternion Ginibre Matrices and their Correlation Functions
• Microscopic correlations of non-Hermitian Dirac operators in three-dimensional QCD
• Non-Hermitean Random Matrix Theory: Summation of Planar Diagrams, the "Single-Ring" Theorem and the Disk-Annulus Phase Transition
• The single ring theorem
• Deviations from the Circular Law

Cited by in corpus (3)

• Recent exact and asymptotic results for products of independent random matrices
• Products of independent elliptic random matrices
• Universal distribution of Lyapunov exponents for products of Ginibre matrices