Products of Independent Quaternion Ginibre Matrices and their Correlation Functions

Abstract

We discuss the product of independent induced quaternion ($\beta =4$) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single Ginibre matrix, but with a more complicated weight function. We find the skew-orthogonal polynomials corresponding to the weight function of the product matrix, and use the method of skew-orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size. The radial behavior of the density of states is studied in the limit of large matrices, and the macroscopic density is discussed. The microscopic limit at the origin, at the edge(s) and in the bulk is also discussed for the radial behavior of the density of states.

19 pages, 2 figures. Typos corrected, references added

References in corpus (5)

• Matrix Models and QCD with Chemical Potential
• On the Asymptotic Spectrum of Products of Independent Random Matrices
• Large N_c confinement and turbulence
• Characteristic polynomials in real Ginibre ensembles
• Non-Hermitean Wishart random matrices (I)

Cited by in corpus (30)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Recent exact and asymptotic results for products of independent random matrices
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Singular value statistics of matrix products with truncated unitary matrices
• Universal microscopic correlation functions for products of truncated unitary matrices
• Relating the Bures measure to the Cauchy two-matrix model
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Products of Independent Gaussian Random Matrices
• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• Determinantal point processes in the plane from products of random matrices
• Statistical limits of dictionary learning: random matrix theory and the spectral replica method
• Probability of all eigenvalues real for products of standard Gaussian matrices
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Asymptotics of finite system Lyapunov exponents for some random matrix ensembles
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Symmetries of the Quaternionic Ginibre Ensemble
• Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Eigenvector statistics of the product of Ginibre matrices
• A note on the limiting mean distribution of singular values for products of two Wishart random matrices
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Diffusion method in Random Matrix Theory
• Wronskian structures of planar symplectic ensembles
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• Time-inhomogeneous random Markov chains
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Truncations of random symplectic unitary matrices
• Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality