Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

Akemann, Gernot; Checinski, Tomasz; Liu, Dang-Zheng; Strahov, Eugene

doi:10.1214/18-aihp888

Cited by 6 publications

(6 citation statements)

References 55 publications

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“…The first case is for a general Pólya ensemble without a shift on either the Hermitian antisymmetric matrices H 1 , the Hermitian matrices H 2 , the Hermitian anti-self-dual matrices H 4 or the complex rectangular matrices M ν . The other two cases we considered are the eigenvalue/squared singular value statistics of the Pólya ensemble added by a either a fixed matrix or a random matrix drawn from a polynomial ensemble on the same space as the Pólya ensemble, see [4,14,38,[41][42][43][44]. All results hold for finite matrix dimension.…”

Section: Discussionmentioning

confidence: 99%

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Kieburg

2019

Random Matrices: Theory Appl.

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Recently subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

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Section: Discussionmentioning

confidence: 99%

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Kieburg

2019

Random Matrices: Theory Appl.

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“…We remark that in Theorem 1 the case of n < N remains unanswered although this is certainly of high interest; see [2,18] and references therein for a relevant question.…”

Section: Statement Of the Problem And Summary Of Resultsmentioning

confidence: 99%

Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

Forrester

Ipsen

Liu

2018

Ann. Henri Poincaré

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We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral. *

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“…will be considered in detail 3 . Here, we are dealing with coupled matrices with the single coupling constant b > 0.…”

Section: )mentioning

confidence: 99%

“…Determinantal processes whose edge behaviour interpolates between the Poisson process and the Airy process were considered in Moshe, Neuberger, and Shapiro [37], and studied further by Johansson [29]. In the context of products of random matrices, a determinantal process with m = 2 interpolating between the classical Bessel-kernel process (at m = 1), and the Meijer G-kernel process for the product of m = 2 independent Gaussian matrices was obtained by the authors in [7,8], and further extended most recently by Liu [33] and a joint work [3]. However, we are not aware of further examples of multilevel interpolating ensembles, with properties described by Theorem 4.1.…”

Section: Introductionmentioning

confidence: 99%

“…Assume that L ≥ n. Then the joint distribution of G 1 and G 2 is given by probability measure (2.1) with a = 1+µ 2µ and b = 1−µ 2µ for 0 < µ < 1, m = 2, V 1 = V 2 = 0, and ν 1 = L − n. The same model with m = 2 and V 1 = V 2 = 0 was generalised in [33,3] to include non-trivial covariance matrices instead of the scalar parameters a and b. Here, again only the squared singular values of the product matrix Y 2 were studied.…”

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

Akemann

Strahov

2018

Ann. Henri Poincaré

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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

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Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

Cited by 6 publications

References 55 publications

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

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

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