Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Liu, Dang-Zheng; Wang, Dong; Zhang, Lun

doi:10.1214/15-aihp696

Cited by 52 publications

(62 citation statements)

References 55 publications

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“…As a consequence of this exact link to a known model, the correlation kernel can be obtained directly from the existing literature. Likewise, the scaling behaviour of the kernel near the origin [33] as well as in the bulk and at the soft edge [34]. It is also worthwhile to mention that our new product (1.14) gives the possibility to construct complex product matrices with half-integer indices {ν i } (albeit interlaced with integer indices), while the rectangular matrices studied by Akemann et al [1] only give rise to integer indices.…”

Section: )mentioning

confidence: 86%

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Forrester

Ipsen

Liu

et al. 2019

Random Matrices: Theory Appl.

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In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and particular Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure X M · · · X 1 (iA)X T 1 · · · X T M , where each X i is a standard real Gaussian matrix, and A is a real anti-symmetric matrix can be determined. For M = 1 and A the bidiagonal anti-symmetric matrix with 1's above the diagonal, this reclaims results of Defosseux. For general M , and this choice of A, or A itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

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Section: )mentioning

confidence: 86%

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Forrester

Ipsen

Liu

et al. 2019

Random Matrices: Theory Appl.

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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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“…See [1] for a survey paper. The bulk and soft edge scaling limits for singular values of Ginibre random matrices are the usual sine and Airy kernels [32], and these classical limits were also established for the Muttalib-Borodin model in the Laguerre case [40].It is natural to expect that the limit (1.5) is not restricted to the case V (x) = x, but holds for much more general external fields. In this paper we consider θ = 1 2 and we show that the hard edge scaling limit (1.5) indeed holds for a large class of external fields V .…”

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confidence: 90%

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Kuijlaars

Leslie

2019

Nonlinearity

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The Muttalib-Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter θ and an external field V . For θ = 1 2 we find the large n behavior of the associated correlation kernel with only few restrictions on V . The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a 3 × 3 Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.with limiting kernel.is Wright's generalization of the Bessel function. For θ = 1, the limit (1.5) reduces to the well-known Bessel kernel in the theory of random matrices. Several other expressions are known for the kernel (1.6). For example, Zhang [40, Theorem 1.2] gives a double contour integral, and if θ or 1/θ is an integer, then (1.5) can be expressed in terms of Meijer G-functions [30]. These so-called Meijer G-kernels appear in singular value distributions for products of random matrices [2, 3] and in their hard edge scaling limit [4,5,23,24,25,31]. See [1] for a survey paper. The bulk and soft edge scaling limits for singular values of Ginibre random matrices are the usual sine and Airy kernels [32], and these classical limits were also established for the Muttalib-Borodin model in the Laguerre case [40].It is natural to expect that the limit (1.5) is not restricted to the case V (x) = x, but holds for much more general external fields. In this paper we consider θ = 1 2 and we show that the hard edge scaling limit (1.5) indeed holds for a large class of external fields V .

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Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Cited by 52 publications

References 55 publications

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

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