Rectangular random matrices, related convolution

Benaych-Georges, Florent

doi:10.1007/s00440-008-0152-z

Cited by 84 publications

(128 citation statements)

References 30 publications

Supporting

Mentioning

123

Contrasting

Order By: Relevance

“…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]. The Markovian structure of free convolution (see [Voi00a] for a basic derivation) was shown in [Voi93] and [Bia98a, Theorem 3.1] to imply the existence of a unique subordination function F : C→C such that…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

View full text Add to dashboard Cite

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.

show abstract

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

View full text Add to dashboard Cite

show abstract

“…This implies that either f (z, H µ⊞ λ ν (z)) takes values also in C \ R + , or it is constant. But it cannot be constant, by equation (4.1) and by the fact that H µ⊞ λ ν (z) is equivalent to z as z tends to zero in C\R + (see [BG1], proposition 4.1). Hence f (z, H µ⊞ λ ν (z)) takes values in C \ R + .…”

Section: The Rectangular Casementioning

confidence: 99%

“…Introduction to the rectangular free convolution and to the related transforms. We recall [BG1] the construction of the rectangular R-transform with ratio λ, and of the rectangular free convolution ⊞ λ with ratio λ, for λ ∈ [0, 1]: one can summarize the different steps of the construction of the rectangular R-transform with ratio λ in the following chain…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

Regularization by Free Additive Convolution, Square and Rectangular Cases

Belinschi

Benaych-Georges

Guionnet

2008

Complex Anal. Oper. Theory

Self Cite

View full text Add to dashboard Cite

Abstract. The free convolution ⊞ is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution ⊞ λ allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups (µt) of probability measures such that µ0 = δ0 and such that for all t > 0 and all probability measure ν, µt⊞ν (or, in the rectangular context, µt⊞ λ ν) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for ⊞, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for µ in a ⊞ λ -continuous semigroup, µ⊞ λ ν either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of µ⊞ λ ν except on a negligible set of points, as well as existence and continuity of a density everywhere.

show abstract

“…Note that the general case m = n is solved similarly with the aid of rectangular spaces, using the theory developed by Benaych-Georges [BG07, BG09a,BG09b]. The paper is organized as follows.…”

mentioning

confidence: 99%

On the asymptotic distribution of block-modified random matrices

Arizmendi

Nechita

Vargas

2015

Journal of Mathematical Physics

View full text Add to dashboard Cite

Abstract. We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of interest in quantum information theory: we unify some known results and we obtain new distributions and various generalizations.

show abstract

Rectangular random matrices, related convolution

Cited by 84 publications

References 30 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Regularization by Free Additive Convolution, Square and Rectangular Cases

On the asymptotic distribution of block-modified random matrices

Contact Info

Product

Resources

About