Limit theorems in free probability theory II

Chistyakov, G. P.; Götze, Friedrich

doi:10.2478/s11533-008-0006-z

Cited by 14 publications

(14 citation statements)

References 17 publications

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“…In fact, defining µ n ∶= (1 − t n )δ 1 + t n δ ζ with arbitrary ζ ∈ T leads to a more general two-parameter family of free Poisson distributions [6,Lemma 6.4], but we shall need only the case ζ = −1 here. For further information on ⊠ including a characterization of ⊠-infinitely divisible distributions and more general limit theorems we refer to [5,7,8,14,15,65,65,64,13].…”

Section: Main Resultmentioning

confidence: 99%

Repeated differentiation and free unitary Poisson process

Kabluchko¹

2021

Preprint

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We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree d trigonometric polynomial are distributed according to some probability measure ν in the large d limit, then the zeroes of its [2td]-th derivative, where t > 0 is fixed, are distributed according to the free multiplicative convolution of ν and the free unitary Poisson distribution with parameter t. In the simplest special case, our result states that the zeroes of the [2td]-th derivative of the trigonometric polynomial (sin θ2 ) 2d (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter t, in the large d limit. The latter distribution is defined in terms of the function ζ = ζt(θ) which solves the implicit equation ζ − t tan ζ = θ and satisfies ζt(θ) = θ + t tan(θ + t tan(θ + t tan(θ + . . .))), Im θ > 0, t > 0.

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Section: Main Resultmentioning

confidence: 99%

Repeated differentiation and free unitary Poisson process

Kabluchko¹

2021

Preprint

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“…Hence the so-called R-transform R of a spectral measure µ a , introduced by Voiculescu in [51], is determined analytically by the inverse function of the Cauchy transform of µ a on the complex plane which is the starting point of the complex analytic theory of the asymptotic approximations of free additive convolution as developed in [23,21,22,24]. Assuming that κ 1 = m 1 = 0, R µa (z) := R a (z) admits a formal inverse power series R (−1) a (z).…”

Section: Non-crossing Partitions In Free Probabilitymentioning

confidence: 99%

Non-crossing partitions

Baumeister¹,

Bux²,

Götze³

et al. 2019

Spectral Structures and Topological Methods in Mathematics

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Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type An, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.

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“…Once we choose a branch, we can associate a Lévy process on T which has the distribution µ ⊛t at time t ≥ 0. For further details see [Céb16,CG08,Par67].…”

Section: Multiplicative Classical Convolutionsmentioning

confidence: 99%

Limit theorems for free Lévy processes

Arizmendi¹

2018

Electron. J. Probab.

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We consider different limit theorems for additive and multiplicative free Lévy processes. The main results are concerned with positive and unitary multiplicative free Lévy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index 1 coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean Lévy processes at small time, obtaining log Boolean stable laws in the limit.

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Limit theorems in free probability theory II

Abstract: Abstract. Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line R + and on the unit circle T we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

Cited by 14 publications

References 17 publications

Repeated differentiation and free unitary Poisson process

Repeated differentiation and free unitary Poisson process

Non-crossing partitions

Limit theorems for free Lévy processes

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