Derivative principles for invariant ensembles

Kieburg, Mario; Zhang, Jiyuan

doi:10.1016/j.aim.2022.108833

Cited by 2 publications

(1 citation statement)

References 23 publications

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“…From a random matrix perspective and at finite matrix size n, the result in [34] provides a bijection between the joint probability density function of the eigenvalues and the one of the singular values, under some assumptions. The related works [33,36,42] bring some tools to exploit this bijection when the singular values are drawn from particular kind of ensembles, such as polynomial ensembles [35,37,41] and, more particularly, for Pólya ensembles, which were formerly coined polynomial ensembles of derivative type [21].…”

Section: Introduction 1state Of the Artmentioning

confidence: 99%

Correlation functions between singular values and eigenvalues

Allard,

Kieburg

2024

Preprint

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Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on j eigenvalues and k singular values that we coin j,k-point correlation measure. We fully derive all j,k-point correlation measures in the simplest cases for matrices of size n = 1 and n = 2 . For n > 2 , we find a general formula for the 1, 1-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. MSC Classification: 60B20 , 15B52 , 43A90 , 42B10 , 42C05

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Section: Introduction 1state Of the Artmentioning

confidence: 99%

Correlation functions between singular values and eigenvalues

Allard,

Kieburg

2024

Preprint

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Projections of orbital measures and quantum marginal problems

Collins

McSwiggen

2023

Trans. Amer. Math. Soc.

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This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn’s problem, the randomized Schur’s problem, and the orbital corners process. In this general setting, we prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to multiplicity problems in representation theory as well as to known results in the symplectic geometry literature. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities.

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Derivative principles for invariant ensembles

Cited by 2 publications

References 23 publications

Correlation functions between singular values and eigenvalues

Correlation functions between singular values and eigenvalues

Projections of orbital measures and quantum marginal problems

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