Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester, Peter J.; Ipsen, Jesper R.

doi:10.1016/j.laa.2016.08.015

Cited by 28 publications

(70 citation statements)

References 70 publications

(121 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As t grows, the eigenvalues increasingly concentrate on the real line. This concentration phenomenon has also been observed for products of other kinds of random matrices [31,32,33,34,35,36,37]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 69%

Time-inhomogeneous random Markov chains

Innocentini

Novaes

2018

J. Stat. Mech.

View full text Add to dashboard Cite

We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, U (t) = Mt · · · M1, with the factors Mi drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of U (t), its largest eigenvalue and its spectrum are obtained exactly for N = 2 and numerically investigated for general N . For large t, the columns are Dirichlet-distributed, however the distribution is different from the initial one. As for the spectrum, we find that the eigenvalues converge to zero exponentially fast and investigate the statistics of the largest stability and Lyapunov exponents, which is are approximated by Gamma distributions. We also observe a concentration of the spectrum on the real line for large t.

show abstract

Section: Introductionmentioning

confidence: 69%

Time-inhomogeneous random Markov chains

Innocentini

Novaes

2018

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…It turns out that the eigenvalues of Y m form a determinantal point process on the complex plane which can be understood as a generalisation of the classical Ginibre ensemble. This fact was first proved in Akemann and Burda [2], see Adhikari, Kishore Reddy, Ram Reddy, and Saha [1], Forrester [21], Ipsen [25], Akemann, Ipsen, and Strahov [6], Forrester and Ipsen [22] for different proofs and extensions of this result. Moreover, it was shown in Akemann, Kieburg, and Wei [4], and in Akemann, Ipsen, and Kieburg [5] that the joint probability density function of the squared singular values of Y m forms a determinantal point process, representing a special polynomial ensemble, a notion introduced later by Kuijlaars and Stivigny [30].…”

Section: Introductionmentioning

confidence: 75%

Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

Akemann

Strahov

2018

Ann. Henri Poincaré

View full text Add to dashboard Cite

Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper we study such processes where the products of up to m complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels.For a special choice of potential, leading to a Gaussian coupling between the mth matrix and the product of all previous m − 1 matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size n. Here, the correlations between the first m − 1 levels equal that of the product of m − 1 independent matrices, whereas all correlations with the mth level are modified. In the hard edge scaling limit at the origin of the spectra of all products we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of m independent complex Gaussian matrices for all levels, including the m-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer G-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level m collapses onto level m − 1, thus leading to the process of m − 1 independent matrices. Finally, in an intermediate regime where the coupling is proportional to n 1 2 , we obtain a family of parameter dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for m = 2. 16 3. Hard edge scaling limits of the multi-level determinantal processes 17 4. The interpolating multi-level determinantal process 20 5. An integration formula for coupled matrices 21 6. Proof of Theorem 2.1 25

show abstract

“…For the matrices X i we follow the strategy used in [15,14] and use a generalised real Schur decomposition . The matrices T i are each strictly upper triangular.…”

Section: The Eigenvalue Pdfmentioning

confidence: 99%

“…where D + = {(x, y) : x 2 + y 2 < 1 and y > 0} denotes the half unit disk. A fundamental feature of (3.1), which follows from the structure of (2.3), is that the corresponding generating function [14,Prop. 5].…”

Section: The Generating Function As a Pfaffian Usementioning

confidence: 99%

“…Our main result, stated in Theorem 2.1, identifies both w r and w c in (1.7). However, as already present in the study of the eigenvalues of the product (1.8) for each X i a real standard Gaussian [14], the expression for w c is too complicated for further analysis (unless m = 1) so we restrict attention to the computation of statistical and probabilistic properties of the real eigenvalues. In Section 3 we give a determinantal formula (with entries given by certain Meijer G-functions) for the probabilities p P m N,k that the product matrix (1.9) has exactly k real eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Forrester

Ipsen

Kumar

2018

Experimental Mathematics

View full text Add to dashboard Cite

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 P m 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 P m 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 formulae, 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.

show abstract

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Cited by 28 publications

References 70 publications

Time-inhomogeneous random Markov chains

Time-inhomogeneous random Markov chains

Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

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

Contact Info

Product

Resources

About