Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Forrester, Peter J.; Wang, Dong

doi:10.1214/17-ejp62

Cited by 62 publications

(90 citation statements)

References 61 publications

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“…These are well defined for all

ν_{i} > - 1

. We know from Kuijlaars and Zhang and from Forrester and Wang (see Eqs. (1), (5), and (8)) that for

θ \in {double-struckZ}_{+}

x^{1 / θ - 1} K^{(c, θ)} false(θ x^{1 / θ}, θ y^{1 / θ} false) = K_{ν_{1}, ..., ν_{θ}} false(x, y false),

where

ν_{j} = \frac{c + j}{θ} - 1, 1 \leq j \leq θ .

Another relevant work on the connection between the two kernels is .…”

Section: M=2 Theory At the Hard Edgementioning

confidence: 81%

Singular Values of Products of Ginibre Random Matrices

Witte

Forrester

2016

Stud Appl Math

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Abstract. The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions 0 F M , also referred to as hyper-Bessel functions. In the case M = 1 it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalised this formalism to general M ≥ 1, but has not exhibited its reduction. After detailing the necessary working in the case M = 1, we consider the problem of reducing the 12 coupled differential equations in the case M = 2 to a single differential equation for the resolvent. An explicit 4-th order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third order nonlinear equation. The small and large s asymptotics of the 4-th order equation are discussed, as is a possible relationship of the M = 2 systems to so-called 4-dimensional Painlevé-type equations.

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“…These are well defined for all

ν_{i} > - 1

. We know from Kuijlaars and Zhang and from Forrester and Wang (see Eqs. (1), (5), and (8)) that for

θ \in {double-struckZ}_{+}

x^{1 / θ - 1} K^{(c, θ)} false(θ x^{1 / θ}, θ y^{1 / θ} false) = K_{ν_{1}, ..., ν_{θ}} false(x, y false),

where

ν_{j} = \frac{c + j}{θ} - 1, 1 \leq j \leq θ .

Another relevant work on the connection between the two kernels is .…”

Section: M=2 Theory At the Hard Edgementioning

confidence: 81%

Singular Values of Products of Ginibre Random Matrices

Witte

Forrester

2016

Stud Appl Math

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“…The last example consists of random matrices for which the joint probability density of eigenvalues is the Muttalib-Borodin Laguerre ensemble [9,26] 1 Such densities can be realized as eigenvalue densities of random matrices, see [1,11,19]. In [19], the authors constructed a random matrix with this eigenvalue density in the following way, in the case where θ is a positive integer and α a non-negative integer.…”

Section: Perturbed Muttalib-borodin Biorthogonal Ensemblesmentioning

confidence: 99%

“…In [19], the authors constructed a random matrix with this eigenvalue density in the following way, in the case where θ is a positive integer and α a non-negative integer. Define α j , j = 1, ..., n by α j = θ(j − 1) + α.…”

Section: Perturbed Muttalib-borodin Biorthogonal Ensemblesmentioning

confidence: 99%

“…The density of the eigenvalues of 1 n X * X is then given by (2.27). The eigenvalue correlation kernel of 1 n X * X can be expressed as [19] 28) with C α a contour enclosing α 1 , ..., α n and C starting at −∞ in the lower half plane, enclosing C α and going back to −∞ in the upper half plane. An alternative expression was given in [37]: It admits the hard edge scaling limit [9,19] lim …”

Section: Perturbed Muttalib-borodin Biorthogonal Ensemblesmentioning

confidence: 99%

“…Finally, we have for z ∈ D δ , 19) as n → ∞, with f n a conformal map defined in a neighbourhood of 0 and satisfying…”

Section: Proof Of Lemma 12mentioning

confidence: 99%

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Gaussian perturbations of hard edge random matrix ensembles

Claeys

Doeraene

2016

Nonlinearity

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We study the eigenvalue correlations of random Hermitian n × n matrices of the form S = M + ǫH, where H is a GUE matrix, ǫ > 0, and M is a positive-definite Hermitian random matrix, independent of H, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of S close to 0 if ǫ tends to 0 together with n → +∞ at a critical speed, depending on the random matrix M . In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our general results to the cases where (i) M is a Laguerre/Wishart random matrix, (ii) M = G * G with G a product of Ginibre matrices, (iii) M = T * T with T a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of M follow a Muttalib-Borodin biorthogonal ensemble.

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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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Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Cited by 62 publications

References 61 publications

Singular Values of Products of Ginibre Random Matrices

Singular Values of Products of Ginibre Random Matrices

Gaussian perturbations of hard edge random matrix ensembles

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

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