Prof. Dr. Roland Speicher
Carlos Vargas Obieta
Oberseminar zur Freien Wahrscheinlichkeit
In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeit.
Zeit und Ort
mittwochs, 14-16 Uhr, SR9 (319)
Vorträge
- 15.10.2014, SR 2 (!) Zheng Zhong (Helsinki).
CLT for product of Wishart random matrices and second order fluctuation.
The linear transfer function x -> H x + z denotes the multiple-input multiple-output (MIMO) wireless communications over consecutive complex Gaussian matrix channels H = Psi Theta. When the matrix dimensions become large, the channel capacity, formulated as the linear spectral statistics of HH', is shown to be asymptotically Gaussian. We give an explicit closed-form expression for the asymptotic variance of channel capacity using second order Cauchy transform.
- 22.10.2014, Octavio Arizmendi (CIMAT, Mexiko).
k-distance graphs of free product of graphs.
In this talk we will explain different products of graphs related to non-commutative notions of independence:
cartesian, star, comb and free. We specialize in the free product and present a recent result in joint work
with Tulio Gaxiola of the spectral distribution of the k-distance graphs of free product of graphs.
- 29.10.2014, Amaury Freslon (Saarbrücken).
An introduction to noncommutative probability for pairs of random variables
We will present a recent idea of Voiculescu to give a meaning to the notion of freeness for pairs of random variables. We will first introduce the basic tools of this bi-freeness theory and then turn to its combinatorial aspects, developped by Chalresworth, Nelson and Skoufranis. Eventually, we will go to the operator-valued setting, where things become more technical.
- 05.11.2014, Amaury Freslon (Saarbrücken).
Operator-valued bi-free probability
This talk follows the one given last week. We will introduce
the operator-valued setting for bi-free probability. The ideas come from
free probability but several technical issues arise, which can be solved
using the combinatorial approach of bi-noncrossing partitions.
- 26.11.2014, 14:00 Uhr s.t. (!) Songzi Li (Toulouse, Frankreich)
On generalized Dyson Brownian motion
In this talk, I will first present the joint work with X.-D.Li and Y.-X Xie on the study of generalized Dyson Brownian motion(GDBM) with a general potential V. Under some reasonable condition on V, we prove the existence and uniqueness of the strong solution to SDE for GDBM; then we study the large N limit behavior of the empirical measure of GDBM; The law of large numbers and the central limit theorem are established.
In the end, I will talk about the Dyson Brownian motion on the octonion algebra: due to the fact that octonions are nonassociative, the dimension of the matrices plays a special role; we give two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent on the law of the spectrum.
- 26.11.2014, 15:00 Uhr s.t. (!) Michael Ulrich (Besancon, Frankreich)
What would a Brownian motion on the Unitary Dual Group (Brown algebra) be?
In Quantum Probability the right object to be able to define Lévy processes are dual groups, as they were defined by Voiculescu. In this talk we will be interested by the Unitary Dual Group U_d, which is generated by d^2 generators u_ij which verify a unitary relation (and which do not commute). We will try to answer to the question of what a Brownian motion on such a structure could be. To do that we will construct a Lévy process on it as limit of a matricial process on U(nd) which will be seen blockwise. We will see that this process verifies nice properties such that we could call it a Brownian Motion on U_d.
- 03.12.2014, Dries Stivigny
(KU Leuven, Belgien)
Products with truncated unitary matrices
Recently, Akemann et al. showed that the squared singular values of products of
complex Ginibre random matrices give rise to a determinantal point process. This
multiplication with a Ginibre matrix was interpreted by Arno Kuijlaars and Dries
Stivigny as a transformation of so called polynomial ensembles. In a recent joint
work with Mario Kieburg and Arno Kuijlaars we showed that multiplication of a
random matrix with a truncation of a Haar distributed unitary matrix can also be
interpreted as a transformation of polynomial ensembles. This formula can be used
repeatedly to describe the joint pdf of the squared singular values of a product of
r truncated unitary matrices and s Ginibre matrices. After this, we will focus on
the global distribution of the squared singular values of these products. In particular
we will introduce an appropriate general class of measures with moments essentially
given by specific Jacobi polynomials with varying parameters. Solving the underlying
moment problem is based on a study of the Riemann surfaces associated to a class of
algebraic equations. This last part is based on joint work with Thorsten Neuschel and
Wolfgang Gawronski.
- 10.12.2014, Mario Diaz
(Queen's University, Kanada)
Random Operator-Valued Matrices
In this talk, we will introduce random operator-valued matrices. Our starting point will be a discussion on how operator-valued free probability may help us to find (good) approximations to information theoretic characteristics of some multiantenna systems. In this context, it is natural to introduce the random version of these operator-valued models (matrices). We will briefly discuss some information theoretic properties of these random operator-valued models and finally we will make some remarks about the kind of tools required to deal with this probabilistic version of operator-valued free probability theory.
- 7.1.2015, Pierre Yves Gaudreau Lamarre
(Ottawa, Kanada)
*-freeness in families of tensor products of non-commutative random variables
In this talk, we will investigate the occurrence of *-freeness in collections $(a_i \otimes b_i : i \in I)$ of tensor products of non-commutative random variables in *-probability spaces with applications in random matrix theory.
- 15.1.2015, 10 Uhr c.t. (!), Seminarraum 9,
Mehmet Madensoy (Saarbrücken)
Noncommutative White Noise Analysis
In this talk we will present selected aspects of the recent works [1] and [2]. More precisely, we will introduce noncommutative analogues of the Kontradiev space of stochastic test functions and Kontradiev space of stochastic distributions, respectively.
We will conclude the talk indicating possible links to (noncommutative) stochastic calculus.
- [1] Daniel Alpay, Palle Jorgensen, Guy Salomon:
On free stochastic processes and their derivatives
- [2] Daniel Alpay, Guy Salomon:
Non-commutative stochastic distributions and applications to
linear systems theory
- 11.2.2015,
Doppelvortrag
14 Uhr s.t. Alexandru Nica (Waterloo)
Star-cumulants of the free unitary Brownian motion
The free unitary Brownian motion (originally considered by
Bercovici-Voiculescu 1992, Biane 1997) is a semigroup of probability
measures on the unit circle, which approach the Haar measure in the
limit when the time $t$ goes to infinity. One of the first significant
calculations of cumulants in free probability (Speicher, 1993) was for
the star-cumulants of the Haar measure on the circle. In this talk
I will present a recent joint work with N. Demni and M. Guay-Paquet
(arXiv:1408.3880) where we discuss the analogous question for
star-cumulants of the free unitary Brownian motion. These star-cumulants
are far more involved than those of the Haar measure, but still turn out
to have tractable features, and offer some interesting combinatorial
puzzles. A significant point is that one can consider their derivative
in the limit $t \to \infty$; this suggests a concept of 'infinitesimal
determining sequence' for R-diagonal distributions, an important class
of non-commutative distributions studied by free probability.
15 Uhr s.t. Adam Skalski (Warschau)
On some rectangular structures related to algebras of functions on free permutation groups
The algebra of functions on the free permutation group S_n^+ is defined as the universal C*-algebra generated by entries of an n by n magic unitary (i.e. a unitary matrix whose entries are orthogonal, not necessarily commuting, projections). It is known to be a source of fascinating combinatorics, and to be connected to several aspects of general C*/von Neumann algebra theory, Hadamard matrices, free independence, etc. In this talk I will discuss two natural generalizations of this algebra, involving rectangular matrices: certain class of homogeneous spaces for S_n^+, and the algebra of functions on the free semigroup of partial permutations of the n points. Based on joint work with Teodor Banica and Piotr Sołtan.
- 12.2.2015, 15 Uhr s.t., Seminarraum 7 (!),
Pierre Tarrago (Paris und Saarbrücken)
Noncommutative symmetric functions and representation theory of free quantum groups
The ring of noncommutative symmetric functions is a free analogue of the usual ring of symmetric functions. I will present in this talk different ways of constructing this ring, and describe the polynomials that appear through these constructions. Then I will explain how this framework may fit into the representation theory of free quantum groups.