On the free convolution with a semi-circular distribution

Biane, Philippe

doi:10.1512/iumj.1997.46.1467

Cited by 173 publications

(274 citation statements)

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“…It is a special case of free convolution (of the law µ and the semi-circular law with variance t) that we shall describe in Chapter 5. A refined study of the analytic properties of free convolution with a semi-circular law, that greatly expands on the results in Lemma 4.3.15, appears in [Bia97b].…”

Section: Bibliographical Notesmentioning

confidence: 95%

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An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 95%

“…A detailed study of free convolution by a semi-circular was done by Biane [Bia97b]. Freeness for rectangular matrices and related free convolution were studied in [BeG06].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…Then, since R µ SC (z) = z and the Cauchy-Stieltjes transform of r i=1 q i δ a i is explicit, one can obtain from (3.7) that the Cauchy-Stiejles transform G of µ SC ⊞ r j=1 q j δ a j is an algebraic function, by performing similar manipulations than in the proof of [7,Lemma 1] and concluding by analytic continuation. More precisely, one obtains that Corollary 3.6.…”

Section: Multiple Hermite Polynomialsmentioning

confidence: 96%

Average characteristic polynomials of determinantal point processes

Hardy¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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We investigate the average characteristic polynomial E N i=1 (z −x i ) where the x i 's are real random variables drawn from a Biorthogonal Ensemble, i.e. a determinantal point process associated with a bounded finite-rank projection operator. For a subclass of Biorthogonal Ensembles, which contains Orthogonal Polynomial Ensembles and (mixed-type) Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as N goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures, and then derive explicit algebraic equations for their Cauchy-Stieltjes transform. AbstractOn s'intéresse au polynôme caractéristique moyen E N i=1 (z −x i ) associéà des variables aléatoires réelles x 1 , . . . , x N qui forment un Ensemble Biorthogonal, c'est-à-dire un processus ponctuel déterminantal associéà un opérateur de projection borné et de rang fini. Pour une sous-classe d'Ensembles Biorthogonaux, qui contient les Ensembles Polynômes Orthogonaux et les Ensembles Polynômes Orthogonaux Multiples (de type mixte), nous obtenons une condition suffisante pour que, presque sûrement, la distribution limite de ses zéros coincide avec la distribution limite des variables aléatoires, quand N tend vers l'infini. De plus, cette condition s'avèreêtreégalement suffisante pour améliorer la convergence en moyenne en convergence presque sûre pour les moments de la mesure empirique associée au processus ponctuel déterminantal. En application, on obtient avec des théorèmes de Voiculescu une description pour les distributions limites des zéros des polynômes d'Hermite et de Laguerre multiples, en termes de convolutions libres de lois classiques avec des mesures atomiques, ainsi que deséquations algébriques explicites pour leurs transformées de Cauchy-Stieltjes.

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“…the measure, whose Stieltjes transform is a solution of (2.18) satisfying (2.12)-(2.14). We refer the reader to works [32,2,4,3] and references therein for a rather complete collection of results on properties of the measure, resulting from the binary operation in the space of the probability measures, defined by a version of system (2.18). This binary operation is called free additive convoluton.…”

Section: Properties Of the Solutionmentioning

confidence: 99%

Law of addition of random matrices

Pastur¹,

Shcherbina²

2011

Eigenvalue Distribution of Large Random Matrices

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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U * n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.

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On the free convolution with a semi-circular distribution

Cited by 173 publications

References 6 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Average characteristic polynomials of determinantal point processes

Law of addition of random matrices

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