Free Jacobi Process

Демни, Низар

doi:10.1007/s10959-007-0110-1

Cited by 39 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. Hence the initial distribution 2μ 0 is the unit mass at θ = 0, and thus the probability distribution of the free Jacobi process with parameter (λ, θ) = (1, 1/2) is exactly that of the free unitary multiplicative Brownian motion via x = cos 2 (θ/2).…”

Section: Analysis Of Probability Distribution Of X Tmentioning

confidence: 69%

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: 69%

Remarks on free mutual information and orbital free entropy

Izumi¹,

Ueda²

2015

Nagoya Math. J.

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“…Next, consider the equation (9) (g κ,t (a)) = 0 and observe that ∂G κ,t ⊂ { (g κ,t (a)) = 0}. Observe also that (g κ,t (a)) = 0 ⇔ (g κ,t (a)) = 0 so that we may restrict (9) to { (a) ≥ 0, (a) ≥ 0}. Now, write g κ,t (a) = 4a 2 (a 2 − 1)e ta (a 2 − κ 2 )(a + 1 + (a − 1)e ta ) 2 = 4 a 2 e ta (|a| 4 + κ 2 − κ 2 a 2 − a 2 )((a + 1) 2 + (a − 1) 2 e 2ta + 2(a 2 − 1)e ta ) |(a 2 − κ 2 )(a + 1 + (a − 1)e ta ) 2 | 2 and substitute a := x + iu, u ≥ 0.…”

Section: Inverse Of the Flow: Analytic Extension Univalence And Alekmentioning

confidence: 99%

“…Now, write g κ,t (a) = 4a 2 (a 2 − 1)e ta (a 2 − κ 2 )(a + 1 + (a − 1)e ta ) 2 = 4 a 2 e ta (|a| 4 + κ 2 − κ 2 a 2 − a 2 )((a + 1) 2 + (a − 1) 2 e 2ta + 2(a 2 − 1)e ta ) |(a 2 − κ 2 )(a + 1 + (a − 1)e ta ) 2 | 2 and substitute a := x + iu, u ≥ 0. Since x = 0 and u = 0 are trivial solutions to (9), then this equation is equivalent to…”

Section: Inverse Of the Flow: Analytic Extension Univalence And Alekmentioning

confidence: 99%

“…Motivated by random matrix theory and especially by the large-size limit of the matrix-valued Jacobi process ( [13]), the first author introduced the free Jacobi process (J t ) t≥0 and performed its free stochastic analysis ( [9]). This is a family of non negative and bounded operators defined as the radial part of the compression of the free unitary Brownian motion by two free (in Voiculescu's sense) orthogonal projections in a finite von Neumann algebra.…”

Section: Introductionmentioning

confidence: 99%

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Inverse of the flow and moments of the free Jacobi process associated with one projection

Демни

Hamdi

2018

Random Matrices: Theory Appl.

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This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we notice that the flow derived in [12] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at z = 1 of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that z = 1 belongs only to the discrete spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse.

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“…The 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}:…”

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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Free Jacobi Process

Cited by 39 publications

References 19 publications

Remarks on free mutual information and orbital free entropy

Remarks on free mutual information and orbital free entropy

Inverse of the flow and moments of the free Jacobi process associated with one projection

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

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