Gaussian perturbations of hard edge random matrix ensembles

Claeys, Tom; Doeraene, Antoine

doi:10.1088/0951-7715/29/11/3385

Cited by 3 publications

(4 citation statements)

References 48 publications

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“…A similar hard edge phase transition occurs in three different regimes for the shifted mean chiral Gaussian ensemble [28]. Recently, some different types of hard-to-soft edge transition have been observed for Gaussian perturbations of hard edge random matrix ensembles by Claeys and Doeraene [16]. Also, see [10] for the famous Baik-Ben Arous-Péché phase transition for largest eigenvalues.…”

Section: 2mentioning

confidence: 70%

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Liu

2017

Constr Approx

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Abstract. Consider the product GX of two rectangular complex random matrices coupled by a constant matrix Ω, where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin's sense, and further that for X being Gaussian the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of ΩΩ * are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel and the finite coupled product kernel associated with GX. In the special case that X is also a Gaussian matrix and Ω is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.

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Section: 2mentioning

confidence: 70%

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Liu

2017

Constr Approx

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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%

“…This result was already derived in [15]. This transformation formula was applied in [14] where the polynomial ensemble was chosen to be the Laguerre ensemble, products of Ginibre matrices or matrices drawn from the Jacobi ensemble and Muttalib-Borodin ensembles.…”

Section: Corollary Iii9 (Simplification Of Theorem Iii8)mentioning

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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“…For the correlation kernels we arrive at the transformation formula which could be useful for asymptotic analysis. A similar formula for the case of a sum with a GUE matrix was given in [11] and it was used for asymptotic analysis in [9] and [10].…”

Section: By Linearity We Havementioning

confidence: 99%

Spherical Functions Approach to Sums of Random Hermitian Matrices

Kuijlaars

Román

2017

International Mathematics Research Notices

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We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair (U(n) ⋉ Herm(n), U(n)). It is inspired by a similar approach of Kieburg and Kösters for products of random matrices. The spherical functions have determinantal expressions because of the Harish-Chandra/ItzyksonZuber integral formula. It leads to remarkably simple expressions for the spherical transform and its inverse. The spherical transform is applied to sums of unitarily invariant random matrices from polynomial ensembles and the subclass of polynomial ensembles of derivative type (in the additive sense), which turns out to be closed under addition. We finally present additional detailed calculations for the sum with a random matrix from a Laguerre Unitary Ensemble.

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Gaussian perturbations of hard edge random matrix ensembles

Cited by 3 publications

References 48 publications

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Spherical Functions Approach to Sums of Random Hermitian Matrices

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