Demonstratio Mathematica, ISSN 0420-1213, 06/2012, Volume 45, Issue 2, pp. 399 - 413

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 12/2019, Volume 100, Issue 3, pp. 987 - 1012

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (goe). In particular, we show how the infinitesimal...

15B52 | 05D40 | 46L54 (primary) | 60B20 (secondary)

15B52 | 05D40 | 46L54 (primary) | 60B20 (secondary)

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 06/2018, Volume 59, Issue 6, p. 63511

We study an extension of Bernstein theorem to the setting of quantum groups. For a d-tuple of free identically distributed random variables, we consider a...

INVARIANCE | ALGEBRAS | FREENESS | FAMILIES | PRODUCT | PHYSICS, MATHEMATICAL | HOPF IMAGES

INVARIANCE | ALGEBRAS | FREENESS | FAMILIES | PRODUCT | PHYSICS, MATHEMATICAL | HOPF IMAGES

Journal Article

Communications in Mathematical Physics, ISSN 0010-3616, 12/2014, Volume 332, Issue 3, pp. 955 - 980

We consider a notion of bi-freeness for systems of non-commutative random variables with two faces, one of left variables and another of right variables. This...

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | COMMUTING RANDOM-VARIABLES | CUMULANTS | ALGEBRAS | MULTIPLICATION | PHYSICS, MATHEMATICAL | Algebra

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | COMMUTING RANDOM-VARIABLES | CUMULANTS | ALGEBRAS | MULTIPLICATION | PHYSICS, MATHEMATICAL | Algebra

Journal Article

Proceedings of the London Mathematical Society, ISSN 0024-6115, 09/2019, Volume 119, Issue 3, pp. 814 - 840

Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in...

16T05 | 16T30 (secondary) | 16T10 | 46L53 | 46L54 (primary) | HOPF-ALGEBRAS | SERIES | THEOREM | DISTRIBUTIONS | MATHEMATICS | TREES | MONOTONE | CUMULANTS | IDEMPOTENTS | BIALGEBRAS | Mathematics | Rings and Algebras

16T05 | 16T30 (secondary) | 16T10 | 46L53 | 46L54 (primary) | HOPF-ALGEBRAS | SERIES | THEOREM | DISTRIBUTIONS | MATHEMATICS | TREES | MONOTONE | CUMULANTS | IDEMPOTENTS | BIALGEBRAS | Mathematics | Rings and Algebras

Journal Article

Inventiones mathematicae, ISSN 1432-1297, 2012, Volume 190, Issue 3, pp. 647 - 697

Given two positive integers n and k and a parameter t∈(0,1), we choose at random a vector subspace V n ⊂ℂ k ⊗ℂ n of dimension N∼tnk. We show that the set of...

15A52 | Mathematics, general | Mathematics | 52A22 | 46L54 | MATHEMATICS | FREE CONVOLUTION | REGULARITY | RANDOM-VARIABLES | ENTROPY

15A52 | Mathematics, general | Mathematics | 52A22 | 46L54 | MATHEMATICS | FREE CONVOLUTION | REGULARITY | RANDOM-VARIABLES | ENTROPY

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 10/2017, Volume 273, Issue 7, pp. 2488 - 2520

Let X1,X2,…,Xn denote i.i.d. centered standard normal random variables, then the law of the sample variance Qn=∑i=1n(Xi−X‾)2 is the χ2-distribution with n−1...

Sample variance | Cancellation of free cumulants | Free infinite divisibility | Wigner semicircle law | MATHEMATICS | CUMULANTS

Sample variance | Cancellation of free cumulants | Free infinite divisibility | Wigner semicircle law | MATHEMATICS | CUMULANTS

Journal Article

Inventiones mathematicae, ISSN 0020-9910, 11/2019, Volume 218, Issue 2, pp. 341 - 411

Since the 1970’s, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of...

Mathematics, general | Mathematics | MATHEMATICS | EQUATIONS | MODULI SPACE | LIMIT | UNITARY | DIMENSION | Physicists | Analysis | Enumeration | Integrals | Mapping | Commutators | Asymptotic methods | Matrix methods | Subgroups

Mathematics, general | Mathematics | MATHEMATICS | EQUATIONS | MODULI SPACE | LIMIT | UNITARY | DIMENSION | Physicists | Analysis | Enumeration | Integrals | Mapping | Commutators | Asymptotic methods | Matrix methods | Subgroups

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 6/2019, Volume 32, Issue 2, pp. 1051 - 1075

We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic...

15B52 | Random matrices | Free probability | Asymptotic expansion | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Spherical integral | 46L54 | STATISTICS & PROBABILITY

15B52 | Random matrices | Free probability | Asymptotic expansion | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Spherical integral | 46L54 | STATISTICS & PROBABILITY

Journal Article

International Mathematics Research Notices, ISSN 1073-7928, 2017, Volume 2017, Issue 18, pp. 5710 - 5750

Easy quantum groups form a class of compact matrix quantum groups which are completely determined by set partitions. They have been defined in 2009 by Banica...

MATHEMATICS | FUSION RULES | FREE WREATH PRODUCT | Operator Algebras | Quantum Algebra | Mathematics

MATHEMATICS | FUSION RULES | FREE WREATH PRODUCT | Operator Algebras | Quantum Algebra | Mathematics

Journal Article

Probability theory and related fields, ISSN 1432-2064, 2012, Volume 157, Issue 3-4, pp. 777 - 815

A fundamental result of Biane (Math Z 227:143–174, 1998) states that a process with freely independent increments has the Markov property, but that there are...

47D06 | Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | Probability Theory and Stochastic Processes | Mathematics | Secondary 60J25 | Primary 46L54 | Quantitative Finance | POLYNOMIALS | FREE BROWNIAN-MOTION | RESPECT | EVOLUTION | MONOTONE | STATISTICS & PROBABILITY | CONVOLUTION | Studies | Probability | Mathematical models | Brownian movements | Brownian motion

47D06 | Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | Probability Theory and Stochastic Processes | Mathematics | Secondary 60J25 | Primary 46L54 | Quantitative Finance | POLYNOMIALS | FREE BROWNIAN-MOTION | RESPECT | EVOLUTION | MONOTONE | STATISTICS & PROBABILITY | CONVOLUTION | Studies | Probability | Mathematical models | Brownian movements | Brownian motion

Journal Article

Journal of theoretical probability, ISSN 1572-9230, 2018, Volume 33, Issue 1, pp. 268 - 294

We study the distribution (with respect to the vacuum state) of a family of partial sums Sm of position operators on weakly monotone Fock space. We show that...

Monotone independence and convolution | Semicircle law | Generalized Catalan recurrences | Non-commutative probability | INDEPENDENCE | 60B99 | STATISTICS & PROBABILITY | 46L53 | 05A18 | 46L54

Monotone independence and convolution | Semicircle law | Generalized Catalan recurrences | Non-commutative probability | INDEPENDENCE | 60B99 | STATISTICS & PROBABILITY | 46L53 | 05A18 | 46L54

Journal Article

Science China Mathematics, ISSN 1674-7283, 5/2019, Volume 62, Issue 5, pp. 921 - 934

Inspired by Speicher's multidimensional free central limit theorem and semicircle families, we prove an infinite dimensional compound Poisson limit theorem in...

multidimensional free Poisson distributions | Mathematics | multidimensional free infinitely divisible distributions | Applications of Mathematics | free probability | 46L54 | MATHEMATICS | MATHEMATICS, APPLIED | FREE PRODUCT

multidimensional free Poisson distributions | Mathematics | multidimensional free infinitely divisible distributions | Applications of Mathematics | free probability | 46L54 | MATHEMATICS | MATHEMATICS, APPLIED | FREE PRODUCT

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 9/2013, Volume 26, Issue 3, pp. 855 - 869

We study the partial transposition W Γ=(id⊗t)W∈M dn (ℂ) of a Wishart matrix W∈M dn (ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mW...

81P45 | Partial transposition | Wishart matrix | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Free Poisson law | 46L54 | STATISTICS & PROBABILITY | Probability | Operator Algebras | Quantum Physics | Physics

81P45 | Partial transposition | Wishart matrix | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Free Poisson law | 46L54 | STATISTICS & PROBABILITY | Probability | Operator Algebras | Quantum Physics | Physics

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 05/2010, Volume 153, Issue 1, pp. 23 - 54

We obtain new Bass-Serre-type rigidity results for III equivalence relations and their von Neumann algebras, coining from free ergodic actions of free products...

MATHEMATICS | REPRESENTATIONS | AMALGAMATED FREE-PRODUCTS | TENSOR-PRODUCTS | MALLEABLE ACTIONS | II1 FACTORS | EQUIVALENCE | Mathematics - Operator Algebras | 46L55 | 46L10 | 46L54

MATHEMATICS | REPRESENTATIONS | AMALGAMATED FREE-PRODUCTS | TENSOR-PRODUCTS | MALLEABLE ACTIONS | II1 FACTORS | EQUIVALENCE | Mathematics - Operator Algebras | 46L55 | 46L10 | 46L54

Journal Article

The Annals of statistics, ISSN 0090-5364, 2016, Volume 44, Issue 2, pp. 598 - 628

The existence of limiting spectral distribution (LSD) of ${\hat \Gamma _u} + \hat \Gamma _u^*$ the symmetric sum of the sample autocovariance matrix ${\hat...

Commutativity | Time series | Eigenvalues | Polynomials | Matrices | Mathematical moments | Mathematical vectors | Random variables | Sampling distributions | Covariance matrices | Free cumulants | Infinite dimensional vector linear process | Semi-circle law | Compound free Poisson | Asymptotically free | Moment method | Wigner matrix | algebra | ID matrix | Non-commutative probability space | Stieltjes transformation | Non-crossing partitions | Limiting spectral distribution | Symmetrized autocovariance matrices | symmetrized autocovariance matrices | moment method | limiting spectral distribution | non-crossing partitions | STATISTICS & PROBABILITY | semi-circle law | compound free Poisson | non-commutative probability space | EMPIRICAL DISTRIBUTION | free cumulants | EIGENVALUES | asymptotically free | TIME-SERIES | DYNAMIC-FACTOR MODEL | COVARIANCE MATRICES | 62M10 | 37M10 | 58C40 | 62J20 | 46L54

Commutativity | Time series | Eigenvalues | Polynomials | Matrices | Mathematical moments | Mathematical vectors | Random variables | Sampling distributions | Covariance matrices | Free cumulants | Infinite dimensional vector linear process | Semi-circle law | Compound free Poisson | Asymptotically free | Moment method | Wigner matrix | algebra | ID matrix | Non-commutative probability space | Stieltjes transformation | Non-crossing partitions | Limiting spectral distribution | Symmetrized autocovariance matrices | symmetrized autocovariance matrices | moment method | limiting spectral distribution | non-crossing partitions | STATISTICS & PROBABILITY | semi-circle law | compound free Poisson | non-commutative probability space | EMPIRICAL DISTRIBUTION | free cumulants | EIGENVALUES | asymptotically free | TIME-SERIES | DYNAMIC-FACTOR MODEL | COVARIANCE MATRICES | 62M10 | 37M10 | 58C40 | 62J20 | 46L54

Journal Article

Complex Analysis and Operator Theory, ISSN 1661-8254, 10/2019, Volume 13, Issue 7, pp. 3023 - 3089

In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to...

Operator Theory | Bi-Boolean independence | Analysis | Mathematics, general | B- $$(\ell , r)$$ ( ℓ , r ) -cumulants | Mathematics | Primary 46L53 | Bi-Boolean partial transforms | Secondary 46L54 | B-$$(\ell , r)$$(ℓ,r)-cumulants | Algebra

Operator Theory | Bi-Boolean independence | Analysis | Mathematics, general | B- $$(\ell , r)$$ ( ℓ , r ) -cumulants | Mathematics | Primary 46L53 | Bi-Boolean partial transforms | Secondary 46L54 | B-$$(\ell , r)$$(ℓ,r)-cumulants | Algebra

Journal Article

Jahresbericht der Deutschen Mathematiker-Vereinigung, ISSN 0012-0456, 3/2017, Volume 119, Issue 1, pp. 3 - 30

This article is an invitation to the world of free probability theory. This theory was introduced by Dan Voiculescu at the beginning of the 1980’s and has...

Random matrix | Non-commutative distribution | Asymptotic representation theory | Mathematics, general | Mathematics | Free probability theory | 46L54

Random matrix | Non-commutative distribution | Asymptotic representation theory | Mathematics, general | Mathematics | Free probability theory | 46L54

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 9/2019, Volume 32, Issue 3, pp. 1220 - 1251

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small...

Perturbation theory | Band matrices | 47A55 | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Wigner matrices | Hilbert transform | 15A52 | Random matrices | Spectral density | 46L54 | BROWNIAN-MOTION | STATISTICS & PROBABILITY | Probability

Perturbation theory | Band matrices | 47A55 | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Wigner matrices | Hilbert transform | 15A52 | Random matrices | Spectral density | 46L54 | BROWNIAN-MOTION | STATISTICS & PROBABILITY | Probability

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 12/2018, Volume 172, Issue 3, pp. 1081 - 1119

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of...

Statistics for Business, Management, Economics, Finance, Insurance | Mathematical and Computational Biology | Full distributions | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Bi-free limit theorem | Operations Research/Decision Theory | Bi-free infinitely divisible distributions | Bi-freely stable distributions | Secondary 60E07 | Primary 46L54 | ANALOG | STATISTICS & PROBABILITY | Geometric probabilities | Distribution (Probability theory) | Analysis | Probabilities | Combinatorial probabilities | Theorems | Convolution | Random variables | Machinery | Probability theory | Combinatorial analysis

Statistics for Business, Management, Economics, Finance, Insurance | Mathematical and Computational Biology | Full distributions | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Quantitative Finance | Bi-free limit theorem | Operations Research/Decision Theory | Bi-free infinitely divisible distributions | Bi-freely stable distributions | Secondary 60E07 | Primary 46L54 | ANALOG | STATISTICS & PROBABILITY | Geometric probabilities | Distribution (Probability theory) | Analysis | Probabilities | Combinatorial probabilities | Theorems | Convolution | Random variables | Machinery | Probability theory | Combinatorial analysis

Journal Article

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