Cyclic independence: Boolean and monotone

Arizmendi, Octavio; Hasebe, Takahiro; Lehner, Franz

doi:10.48550/arxiv.2204.00072

Cited by 2 publications

(9 citation statements)

References 17 publications

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“…It is quite interesting that the inverse Markov-Krein transform appears in the calculations of infinitesimal distributions. 4 This was already observed in [26] for the single variate case. In the present paper, we propose a notion of a "multivariate inverse Markov-Krein transform.…”

Section: Overview Of Main Results and Structure Of The Papersupporting

confidence: 59%

“…In 2018, Collins, Hasebe, and Sakuma gave an abstract framework to Shlyakhtenko's aforementioned work and named the calculation rule of moments cyclic-monotone independence because of its resemblance with monotone independence [20]. Not only being similar, cyclic-monotone independence is directly connected to monotone independence: Arizmendi, Hasebe and Lehner proved that the canonical operator model for monotone independence on the tensor product Hilbert space satisfies cyclic-monotone independence with respect to the vacuum state and the trace [4]; Cébron, Dahlqvist, and Gabriel showed how monotone independence appears from cyclic-monotone independence in an abstract setting [17]; Collins, Leid and Sakuma constructed matrix models for monotone independence and cyclic-monotone independence [21].…”

Section: Backgroundsmentioning

confidence: 99%

“…Other models of cyclic-(anti)monotone independence are provided by Collins, Leid and Sakuma [21] and Arizmendi, Hasebe and Lehner [4]. Notably, the definition of cyclic-monotone independence in [4] was given for any number of subalgebras, not limited to two subalgebras. However, in the present paper we only discuss cyclic-antimonotone independence for a pair of subalgebras.…”

Section: Distributions and Convolutionsmentioning

confidence: 99%

“…(a) Our definition is more or less the same as that of [17,Definition 2.15], but is different from [4,Definition 7.2] because we select the factor ϕpa 0 a n q instead of ϕpa n a 0 q in (2.1). Of course, this does not matter if ϕ is tracial, which is assumed in [20,21].…”

Section: Distributions and Convolutionsmentioning

confidence: 99%

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The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

Fujie¹

2022

ALEA

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We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then the difference of rescaled EEDs of the whole matrix and of its principal submatrix concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of m. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.

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Section: Overview Of Main Results and Structure Of The Papersupporting

confidence: 59%

Section: Backgroundsmentioning

confidence: 99%

Section: Distributions and Convolutionsmentioning

confidence: 99%

Section: Distributions and Convolutionsmentioning

confidence: 99%

See 2 more Smart Citations

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

Fujie¹

2022

ALEA

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“…The cyclic monotone independence was introduced in [CHS18] as a replacement of monotone independence in order to describe the behaviour of some random matrices with respect to the non-normalized trace (see also the recent [AHL22] for an alternate point of view on cyclic monotone independence).…”

Section: Monotone Cyclic Independencementioning

confidence: 99%

Freeness of type $B$ and conditional freeness for random matrices

Cébron¹,

Dahlqvist²,

Gabriel³

2022

Preprint

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The asymptotic freeness of independent unitarily invariant N × N random matrices holds in expectation up to O(N −2 ). An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type B. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type B, conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles. * G.C. has been partly supported by the ERC advanced grant "Noncommutative distributions in free probability", partly by the Project MESA (ANR-18-CE40-006) and partly by the Project STARS (ANR-20-CE40-0008) of the French National Research Agency. G.C. wishes to thank Roland Speicher, for giving him the opportunity to co-organize the Mini-workshop Notions of freeness in April 2015, during which the authors conceived of the first ideas that led to the present work.† F.G. has been partly supported by the ERC SG CONSTAMIS.

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Cyclic independence: Boolean and monotone

Cited by 2 publications

References 17 publications

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

Freeness of type $B$ and conditional freeness for random matrices

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