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

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

doi:10.48550/arxiv.1412.6777

Cited by 20 publications

(39 citation statements)

References 34 publications

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“…For the rearrangement of the matrices inside the superdeterminant we have employed the identities Sdet (AB) = Sdet (A) Sdet (B) and Sdet (H ⊗ 1 k|k ) = 1 (43) for any two square supermatrices A and B and any ordinary square matrix H. Next, we apply (38) to (41) and obtain…”

Section: Supersymmetry Methodsmentioning

confidence: 99%

“…Indeed in Ref. [43], those local spectral statistics have been proven for this kind of random matrix. However the tail statistics do not share this behaviour.…”

Section: Introductionmentioning

confidence: 90%

“…In Ref. [43], those have been proven to be those shared with the GUE. For instance, the soft edge lies at λ min = 1/4 in the macroscopic scaling for any L ∈ N. Thus, we should find the microscopic level density [1,2] ρ Airy (λ) = lim…”

Section: Numerical Observationsmentioning

confidence: 98%

“…In our present work we consider two specific random matrix ensembles. The first one is about the singular values of a product of complex inverse Ginibre matrices, see [40,41,42,43] for an analytical computation of the finite N (matrix dimension) statistics as well as the hard edge statistics. This ensemble is not stable for finite matrix dimension, it is known [44,45] that the limiting macroscopic level density is stable under free convolution (sum of identical and independent copies of the random matrix in the large N limit).…”

Section: Introductionmentioning

confidence: 99%

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Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Kieburg,

Monteleone

2021

Preprint

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We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poissonlike statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

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Section: Supersymmetry Methodsmentioning

confidence: 99%

“…Indeed in Ref. [43], those local spectral statistics have been proven for this kind of random matrix. However the tail statistics do not share this behaviour.…”

Section: Introductionmentioning

confidence: 90%

Section: Numerical Observationsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Kieburg,

Monteleone

2021

Preprint

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“…Particularly, we consider unitarily invariant Hermitian random matrices with heavy tails. Well-known heavy tailed unitarily invariant random matrices are the Cauchy ensemble [83,79], the Cauchy-Lorentz ensemble [82] (also known as the matrix Student t-function in statistics [45,11,77]), and the Hermitised products of Ginibre with inverse Ginibre matirces [2,33,58]. The latter are only one kind of several multiplicative Pólya ensembles [52,53,34] that can exhibit heavy tails.…”

mentioning

confidence: 99%

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

Kieburg¹,

Zhang²

2021

Preprint

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We consider random matrix ensembles on the Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and Pólya ensembles. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions.

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Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

Forrester

Liu

2015

Commun. Math. Phys.

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The singular values squared of the random matrix product Y = GrG r−1 · · · G 1 (G 0 + A), where each G j is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A * A are equal to bN , the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of 0 < b < 1 is independent of b, and is in fact the same as that known for the case b = 0 due to Kuijlaars and Zhang. The critical regime of b = 1 allows for a double scaling limit by choosing b = (1 − τ / √ N ) −1 , and for this the critical kernel and outlier phenomenon are established. In the simplest case r = 0, which is closely related to nonintersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of b > 1 with two distinct scaling rates. Similar results also hold true for the random matrix product TrT r−1 · · · T 1 (G 0 +A), with each T j being a truncated unitary matrix.

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

Cited by 20 publications

References 34 publications

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

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