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.

Santosh Kumar

Physics

(SoNS) School of Natural Sciences

Forrester P.J.

Fulltext

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Shiv Nadar University

Journal of Theoretical Probability

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean. © 2020 IOP Publishing Ltd.

Journal of Physics A: Mathematical and Theoretical

Spectral statistics for the difference of two Wishart matrices

<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?

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.

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

Random Matrices: Theory and Applications

Central limit theorems for the real eigenvalues of large Gaussian random matrices

Linear Algebra and its Applications

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

The Annals of Applied Probability

What is the probability that a large random matrix has no real eigenvalues?

Journal of Statistical Physics

The Real Ginibre Ensemble with k = O ( n ) Real Eigenvalues

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