Local Universality in Biorthogonal Laguerre Ensembles

Zhang, Lun

doi:10.1007/s10955-015-1353-3

Cited by 29 publications

(31 citation statements)

References 56 publications

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“…Moreover, the M = 1 case is the so-called Laguerre Muttalib-Borodin ensemble for which the biorthogonal polynomials are known. In particularly, we know from [30,40,13] that…”

Section: Initial Matrix As An Elementary Anti-symmetric Matrix and Prmentioning

confidence: 99%

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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“…Moreover, the M = 1 case is the so-called Laguerre Muttalib-Borodin ensemble for which the biorthogonal polynomials are known. In particularly, we know from [30,40,13] that…”

Section: Initial Matrix As An Elementary Anti-symmetric Matrix and Prmentioning

confidence: 99%

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Forrester

Ipsen

Liu

et al. 2019

Random Matrices: Theory Appl.

Self Cite

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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: 91%

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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“…and then the right hand side of (1.11) is a well-known (see e.g. [43]) integral representation of the hard edge Bessel kernel. A slightly stronger version of (1.9) allows one to show that the large n limit of the smallest particle distribution is given by…”

Section: Introductionmentioning

confidence: 99%

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Claeys

Girotti

Stivigny

2017

International Mathematics Research Notices

134

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We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2×2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

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Local Universality in Biorthogonal Laguerre Ensembles

Cited by 29 publications

References 56 publications

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

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

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

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