We provide a proof to a recent conjecture by Forrester (2014 J. Phys. A: Math. Theor. 47 065202) regarding the algebraic and arithmetic structure of Meijer G-functions which appear in the expression for probability of all eigenvalues real for the product of two real Gaussian matrices. In the process we come across several interesting identities involving Meijer G-functions. © 2015 IOP Publishing Ltd.

Santosh Kumar

Physics

(SoNS) School of Natural Sciences

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Shiv Nadar University

Journal of Physics A: Mathematical and Theoretical

<p>A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product Pm of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of Pm form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulas, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity. © 2018 Taylor &amp; Francis.</p>

Experimental Mathematics

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

The probability that all eigenvalues of a product of m independent N× N subblocks of a Haar distributed random real orthogonal matrix of size (Li+ N) × (Li+ N) , (i= 1 , ⋯ , m) are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any m and with each Li even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases. © 2017, Springer Science+Business Media New York.

Journal of Theoretical Probability

The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices

We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andréief's integration formula, is used to compactly express the r-point correlation function of eigenvalues. This representation circumvents the complications encountered in the usual approaches, and the answer is obtained immediately by examining the joint density of eigenvalues. We validate our analytical results using Monte Carlo simulations. © 2015 American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure

International Mathematics Research Notices

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

IEEE Transactions on Communications

On the Outage Capacity of Orthogonal Space-Time Block Codes Over Multi-Cluster Scattering MIMO Channels

Acta Physica Polonica B

Recent Exact and Asymptotic Results for Products of Independent Random Matrices

Communications in Mathematical Physics

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices