Free products of large random matrices - a short review of recent developments

Abstract

We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use these methods in practice. In particular we calculate eigenvalue densities of products of Gaussian Hermitian and non-Hermitian matrices including combinations of GUE and Ginibre matrices.

Presented at the workshop: Inference, Computation, and Spin Glasses, Sapporo, July 28th-30th 2013

References in corpus (5)

• Singular value correlation functions for products of Wishart random matrices
• Universal microscopic correlation functions for products of independent Ginibre matrices
• New spectral relations between products and powers of isotropic random matrices
• Eigenvalue density of Wilson loops in 2D SU(N) YM
• Commutative law for products of infinitely large isotropic random matrices

Cited by in corpus (13)

• Rotational invariant estimator for general noisy matrices
• Relating the Bures measure to the Cauchy two-matrix model
• Exact Relation between Singular Value and Eigenvalue Statistics
• Products of Random Matrices from Polynomial Ensembles
• Composition Dependent Instabilities in Mixtures With Many Components
• Eigenvector statistics of the product of Ginibre matrices
• Inferring hidden states in Langevin dynamics on large networks: Average case performance
• Statistical Description of Transport in Multimode Fibers with Mode-Dependent Loss
• Cleaning large-dimensional covariance matrices for correlated samples
• Spectral statistics for the difference of two Wishart matrices
• Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
• Time-inhomogeneous random Markov chains
• Invariant sums of random matrices and the onset of level repulsion