Prof. Dr. Roland Speicher
Octavio Arizmendi Echegaray
Oberseminar zur Freien Wahrscheinlichkeit
In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeit.Zeit und Ort
mittwochs, 16-18 Uhr, SR6 (216)Vorträge
- 19.10.2011 Victor Perez Abreu
On an infinitely divisible Gamma random matrix
A new example of infinitely divisible positive definite Gamma random matrix is introduced.
It has properties which make it appealing for modeling under an infinite divisibility framework.
Relations with the Wishart random matrix are pointed out. Finally we will present some results
on the asymptotic spectral distribution of this Gamma random matrix. This is joint work in
progress with Robert Stelzer.
- 26.10.2011 Octavio Arizmendi Echegaray
Free subordinators and the square of a symmetric free infinitely divisible distributions
In this talk we will treat the class of freely infinitely divisible distributions corresponding to
the classical infinitely divisible distributions with positive support: the free regular infinitely
divisible measures. We clarify many descriptions of this class, including their role in free levy
processes: they correspond to free subordinators. We will show that the square of a symmetric
free infinitely divisible distribution is also freely infinitely divisible. Moreover it can be represented
as the free multiplicative convolution of a free poisson and a free regular measure. This gives two
new explicit examples of measures which are infinitely divisible with respect to both classical and
free convolutions: $ \chi^2$ and $ F(1,1)$.
We also prove that the class of free regular measures is closed under free multiplicative convolution.
Furthermore, we show that other operations in non-commutative probability preserve free regular
infinite divisibility such as $t$th Boolean power for $0\leq t\leq 1$, the $t$th free multiplicative
power for $t\geq 1 $ and weak convergence.
Finally, we give some examples and conjectures regarding free regular infinite divisibility.
- 30.11.2011 Michael Stolz
Stein's method and the multivariate CLT for traces of powers on the classical compact groups
Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups,
distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large
matrix size $n$, the vector $Tr(M_n), Tr(M_n^2), \ldots, Tr(M_n^d))$ tends to a vector of independent
(real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single
power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method
of exchangeable pairs.
In this talk, I will give a leisurely introduction to Stein's method for normal approximation and explain
a multivariate version of Fulman's result (joint work with Christian Döbler, to appear in EJP).
- Dienstag (!), 10.01.2012, 16 Uhr ct, HS IV Claus Köstler
Ein operatoralgebraischer Zugang zur Darstellungstheorie der unendlichen symmetrischen Gruppe
Die unendliche symmetrische Gruppe ist ein typisches Beispiel fuer eine "grosse Gruppe", die keine
(nicht-trivialen) irreduziblen Darstellungen besitzt. Thoma bestimmte erstmals in den 60er Jahren ihre
extremalen Charaktere. Alternative Beweise dieses Satzes von Thoma wurden von Vershik & Kerov
in den 80er Jahren, sowie von Okounkov in den 90er Jahren gegeben. Ich werde in meinem Vortrag
einen neuen operatoralgebraischen Beweis des Satzes von Thoma vorstellen.
- 01.02.2012 Natasha Blitvic
The (q,t)-Gaussian Process
The setting for this talk is the deformed commutation relations and Fock space representations of deformed
quantum harmonic oscillator algebras. Building on the work of Bozejko and Speicher in the single-parameter
case, we will begin by constructing the (q,t)-Fock space, a two-parameter deformation of the Bosonic and
Fermionic Fock spaces. We will discuss the probablistic interpretation of the algebras of bounded linear
operators on this space; that is, the ensuing two-parameter continuum of non-commutative probability theories,
with particular emphasis on the operators playing the role of the classical Gaussian random variables.
We will focus on the remarkable combinatorial structure underlying these objects and, if time permits, their
intimate ties to other mathematical entities.
- 08.02.2012 Tobias Mai
Berry-Esseen estimates for a multivariate free central limit theorem
We address the question of a Berry-Esseen type theorem for the speed of convergence in a multivariate free
central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the
normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting
operator-valued semicircular element. Since we have to deal with non-self-adjoint operators in general, we
introduce the notion of matrix-valued resolvent sets and we study the behavior of Cauchy transforms on them.
- 17.02.2012 Octavio Arizmendi
Free and Boolean Strictly Stable Laws revisited.
Infinitely divisible measures play an important role in probability theory, since they arise from very general
limit theorems. An important subclass of them is the one of stable laws.
Bercovici and Pata showed that classical, Free and Boolean strictly stable laws are in bijection. In this talk
we consider different aspects of Free and Boolean Stable laws.
We start by proving that the Cauchy distribution is the only fixed point of the Boolean-to-Free Bercovici Pata
Bijection. Next, we show a reproducing property for all free and Boolean stable laws. This generalizes previous
results by Biane, for positive free stable laws and by Arizmendi and Perez Abreu, for symmetric ones.
Moreover, this reproducing property implies the free infinite divisibility of Boolean Stable laws, with index
1/n, for n integer greater than 2.
Belinschi and Nica introduced a composition semigroup on the set of probability measures B_t. This semigroup
coincides with the Boolean-to-Free Bercovici Pata Bijection, for t=1.Using this semigroup, they introduced a
free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible
or not.
Finally, we address the problem of calculating the free divisibility indicator of Boolean and free stable laws.
In particular we find new examples of probability measures with free divisibility indicator infinity.
This is a joint work with Takahiro Hasebe.
- 29.02.2012 Uwe Franz
What are Brownian motions on compact quantum groups.
We recall the definition of Levy processes on quantum groups and discuss the problem of characterising Brownian
motions. We suggest generalisations to compact quantum groups of several desirable properties:
gaussianity, symmetry, and invariance under the adjoint action.
While some compact quantum groups, e.g., the Woronowicz quantum group SU_q(2) with q^2\not=1 and the other
q-deformations of compact simple Lie groups, have no quantum stochastic processes satisfying all these properties,
it turns out that the liberated orthogonal quantum group O_n^+ has a unique Brownian motion.
- 29.02.2012 Sören Möller
An overview of (complete) approximation properties for C*-algebras
There are a number of approximation properties a C*-algebra might have. This talk will give an overview over some
of them, focusing on implications between properties as well as the special case of group algebras, in which many
of these properties can be translated to thegroups.
- 14.03.2012 James Mingo
The Moebius Function and the Weingarten Function
Two of the most important functions in free probability are the Moebius function and the Weingarten function.
The Moebius function is used to write cumulants in terms of moments and the Weingarten function is used
to calculate integrals against the Haar measure of the unitary group U(n).
I will explain the background to these functions and show how they are related by a simple equation.
This is joint work with Roland Speicher.
- 14.03.2012 Emily Redelmeier
A Graphical Calculus for Haar-Distributed Orthogonal Matrices and Second-Order Real Freeness
We present a graphical calculus for computing the expected values of traces of expressions involving
Haar-distributed orthogonal matrices, similar to the genus expansion for various Gaussian matrices.
As with the real genus expansions, the expansion includes nonorientable surfaces, and the order of
a term depends only on its Euler characteristic.
We use this to demonstrate that independent matrices which are orthogonally in general position are
second-order real free.
As a corollary, any orthogonally invariant distribution, including the Haar-distributed matrices themselves,
are second-order real free.
- 23.03.2012 Octavio Arizmendi
Second order even and R-diagonal operators.
We propose definitions of even and R-diagonal elements of second order, including the semicircular and haar unitaries considered by Mingo and Speicher in connection of fluctuation of Gaussian Unitary Ensembles and Haar distributed random unitaries, respectively. We give a formula for the (2nd order) free cumulants of the square x^2 of a second order even element in terms of the (2nd order) free cumulants of x. We get similar results the free cumulants of aa* when a is second order R-diagonal. We also show that if a is R-diagonal and b is second order free from a then ab is also R-diagonal, showing that second order R-diagonal operators exist in abundance.
This is a joint work with James Mingo.
Aktualisiert am: 14. März 2012 Octavio Arizmendi
