Recent exact and asymptotic results for products of independent random matrices

Abstract

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.

39 pages, review based on presentations at "Random Matrix Theory: Foundations and Applications", July 1-6, 2014, Krakow, Poland; v2: typo corrected, 2 clarifying comments added

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• Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis
• Random Matrix Theory and Quantum Chromodynamics
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• Symmetries of the Quaternionic Ginibre Ensemble
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• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• May-Wigner transition in large random dynamical systems
• Spectral statistics for the difference of two Wishart matrices
• Singular values for products of two coupled random matrices: hard edge phase transition
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
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• Products of Complex Rectangular and Hermitian Random Matrices
• Supersymmetry for Products of Random Matrices
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• Correlation kernels for sums and products of random matrices
• Matrix integrals and Hurwitz numbers
• Time-inhomogeneous random Markov chains
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Gaussian perturbations of hard edge random matrix ensembles
• Large n limit for the product of two coupled random matrices
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• Integrable structure of products of finite complex Ginibre random matrices
• Lyapunov exponents and eigenvalues of products of random matrices
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
• Phase transitions for infinite products of large non-Hermitian random matrices
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Integrals of tau functions
• On eigenvalue distributions of large auto-covariance matrices
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• The Product of Gaussian Matrices is Close to Gaussian
• Real eigenvalue statistics for products of asymmetric real Gaussian matrices
• Entropy and singular-value moments of products of truncated random unitary matrices
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• How many eigenvalues of a product of truncated orthogonal matrices are real?
• Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles
• Random Matrix Time Series