Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices

Abstract

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product $X_M\cdots X_2{X}_{1}A{X}_1^T{X}_2^T\cdots X_M^T$, where each $X_i$ is a standard real complex Gaussian matrix, and $A$ is real anti-symmetric shares exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the $X_i$ are chosen as sub-blocks of Haar distributed real orthogonal matrices.

32 pages

References in corpus (12)

• Recent exact and asymptotic results for products of independent random matrices
• Singular value statistics of matrix products with truncated unitary matrices
• Truncations of Random Orthogonal Matrices
• Energy correlations for a random matrix model of disordered bosons
• Exact Relation between Singular Value and Eigenvalue Statistics
• Products of Random Matrices from Polynomial Ensembles
• Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• Polynomial Ensembles and Pólya Frequency Functions
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Additive Matrix Convolutions of Pólya Ensembles and Polynomial Ensembles

Cited by in corpus (12)

• The Page Curve for Fermionic Gaussian States
• Harmonic analysis for rank-1 Randomised Horn Problems
• Second-order statistics of fermionic Gaussian states
• Entropy fluctuation formulas of fermionic Gaussian states
• Spectral statistics for the difference of two Wishart matrices
• Derivative principles for invariant ensembles
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• Entanglement capacity of fermionic Gaussian states
• Volume-law entanglement entropy of typical pure quantum states
• Average mutual information for random fermionic Gaussian quantum states