Spectral distribution of the free Jacobi process

Демни, Низар; Hamdi, Tarek; Hmidi, Taoufik

doi:10.1512/iumj.2012.61.5034

Cited by 26 publications

(53 citation statements)

References 13 publications

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“…(2) The above proposition also recaptures, as its specialization, the main theorem of [9]. In fact, the free Jacobi process with parameter (λ, θ) = (1, 1/2) [7] is exactly our X t (viewed as a random variable in (QMQ, 1 τ (Q) τ )) with P = Q and τ (P ) = τ (Q) = 1/2.…”

Section: Analysis Of Probability Distribution Of X Tmentioning

confidence: 65%

Remarks on free mutual information and orbital free entropy

Izumi¹,

Ueda²

2015

Nagoya Math. J.

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Section: Analysis Of Probability Distribution Of X Tmentioning

confidence: 65%

Remarks on free mutual information and orbital free entropy

Izumi¹,

Ueda²

2015

Nagoya Math. J.

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“…When J t is considered as a positive operator valued in the compressed algebra (P A P, τ /τ (P )), its spectral distribution µ t 1 is a probability distribution supported in [0, 1] and the positive real number τ (P )µ t {1} encodes the general position property for {P, Y t QY * t } ( [4], [13]). On the other hand, the couple of papers [7] and [8] aim to determine the Lebesgue decomposition of µ t when both projections coincide P = Q. In particular, when τ (P ) = 1/2, a complete description was given in [7] (see Corollary 3.3 there and [13] for another proof): at any time t ≥ 0, µ t coincides with the spectral distribution of Y 2t + Y * 2t + 21 4 considered as a positive operator in (A , τ ) (the spectral distribution of Y 2t , say η 2t , was described in [2], Proposition 10).…”

Section: Reminder and Motivationmentioning

confidence: 99%

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

Демни

2015

Complex Anal. Oper. Theory

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Abstract. We pursue the study started in [8] of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around z = 0 of the flow determined in [8]. When the rank of the projection equals 1/2, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in (0, 1), we derive a contour integral representation for the first derivative of the Taylor series which is a major step toward the analytic extension of the flow in the open unit disc. Reminder and motivationThe free Jacobi process (J t ) t≥0 was introduced in [6] as the large-size limit of the Hermitian matrix Jacobi process ([9]). It is built as the radial part of the compression of the free unitary Brownian motion (Y t ) t≥0 ([1]) by two orthogonal projections {P, Q}:In this definition, both families of operators {P, Q} and (Y t ) t≥0 are * -free (in Voiculescu's sense) in a von Neumann algebra A endowed with a finite trace τ and a unit 1. When J t is considered as a positive operator valued in the compressed algebra (P A P, τ /τ (P )), its spectral distribution µ t 1 is a probability distribution supported in [0, 1] and the positive real number τ (P )µ t {1} encodes the general position property for {P, Y t QY * t } ([4], [13]). On the other hand, the couple of papers [7] and [8] aim to determine the Lebesgue decomposition of µ t when both projections coincide P = Q. In particular, when τ (P ) = 1/2, a complete description was given in [7] (see Corollary 3.3 there and [13] for another proof): at any time t ≥ 0, µ t coincides with the spectral distribution of Y 2t + Y * 2t + 21 4 considered as a positive operator in (A , τ ) (the spectral distribution of Y 2t , say η 2t , was described in [2], Proposition 10). For an arbitrary rank τ (P ) ∈ (0, 1), only the discrete part in the Lebesgue decomposition of µ t was determined in [8] (see Theorem 1.1). As to its absolutely-continuous part with respect to Lebesgue measure in [0,1], it was related to that of the spectral distribution, say ν t , of the unitary operatorActually, the density of the former distribution is related to the density of the latter through the Caratheodory extension of the Riemann map of the cut planeThe key ingredient leading to this partial description is a flow ψ t so far defined and exploited in an interval of the form (−1, z t ) for some z t ∈ (0, 1), t > 0. When τ (P ) = 1/2, ψ t is a one-to-one map from a Jordan domain onto the open unit disc D and its compositional inverse coincides, up to a Cayley transform, with the Herglotz transform of η 2t . For arbitrary ranks, ψ t is locally invertible and further information on ν t (which in turn provide information on µ t ) necessitates the investigation of the analytic extension of of the local inverse of ψ t in D. Moreover, it was shown in [8] that ν t converges weakly to a probability measure ν ∞ whose support disconnects as ...

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“…One may now consider the family of operator L (λ ) defined through 9) which is symmetric with respect to the measure µ λ = D(Z,Z) (2λ −5)/6 dZ, with support the set {D(Z,Z) ≥ 0} (where dZ is a short hand for the Lebesgue measure in the complex plane) as a direct (although a bit tedious) computation shows from a direct application of formula (1.5) (see Section 1.4 for a proof in a general context which applies in particular here).…”

Section: If We Write Zmentioning

confidence: 99%

h-Transforms and Orthogonal Polynomials

Bakry

Zribi

2015

Lecture Notes in Mathematics

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We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition. Key words: Symmetric diffusions, orthogonal polynomials, h-transformsMSC classification : 33C52, 31C35, 35K05, 60J60, 60J45. 1March 25, 2015Chapter 1 h-transforms and orthogonal polynomialsDominique Bakry and Olfa Zribi IntroductionWhen the first author of this paper was a student, he was attending the DEA course of Marc Yor, about Brownian motions and the many laws that one would compute explicitly for various transformations on the trajectories. It looked like magic, and was indeed. In particular, the fact that conditioning a real Brownian motion to remain positive would turn it into a Bessel process in dimension 3, that is the norm of a 3-dimensional Brownian motion, seemed miraculous. Of course, there are much more striking identities concerning the laws of Brownian motion that one may find in the numerous papers or books of Marc Yor (see [26] for a large collection of such examples). The same kind of conditioning appears in many similar situations, and specially in geometric models. This is related to the fact that we then have explicit h (or Doob)-transforms.This relation between conditioning and h-transform was first put forward by J.L. Doob [11], and is described in full generality in Doob's book [12]. However, this kind of conditioning has been extended in various contexts, and very reader friendly explained by Marc Yor and his co-authors, in particular in [29,27]. The fact that conditioning a d-dimensional model to remain in some set produces a new model in the same family (whatever the meaning of "family"), moreover with dimension d + 2, appears to be a general feature worth to be further understood. It turns out that the most known models have a common explanation, due to an underlying structure related to orthogonal polynomials. The scope of this short note is to shed light on these connections. The paper is organized as follows. In Section 1.2, we present the langage of symmetric diffusion operators that we shall use in the core of the text, and explain what h-transforms are. Section 1.3 gives a few classical and known examples (some of them less well known indeed). They all follow the same scheme, explained in Section 1.4, which provides the general framework, related to the study of orthogonal polynomials which are eigenvectors of diffusion operators. The last Section 1.5 provides further examples, as applications of the main result, inspired from random matrix theory. Symmetric diffusion operators, images and and h-transforms Symmetric diffusion operatorsWe give here a brief account of the tools and notations that we shall be using throughout this paper, most of them following the general setting described in [2]. A symmetric diffusion process (ξ t ) on a measurable space E may be described by its generator L , acting on a good algebra A of real valued funct...

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Spectral distribution of the free Jacobi process

Cited by 26 publications

References 13 publications

Remarks on free mutual information and orbital free entropy

Remarks on free mutual information and orbital free entropy

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

h-Transforms and Orthogonal Polynomials

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