Prof. Dr. Roland Speicher
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
- 22.4.2015, Dominik Janzing (Tübingen)
Free Independence in Causal Inference
Recently proposed methods for causal inference
infer whether 'X causes Y' and 'Y causes X'
after observing P(X,Y) using the following postulate:
whenever X causes Y, then P(X) contains no information about P(Y|X) and
vice versa, in a sense to be defined.
I will sketch different attempts to define this 'independence', one of
it uses independence of free probabilitity theory: If X and Y are
high-dimensional variables then the covarariance matrix of X and the
structure matrix relating X and Y should be free independent. I will
also mention a method to detect hidden common causes that is under
development which seems to be related to free probability theory in a way that is
not completely clear yet.
- 6.5.2015, Guillaume Cébron (Saarbrücken)
Haar states on the unitary dual group
The *-algebra generated by the non-commuting coefficients of a unitary matrix is often called the « unitary dual group » because it has a structure of dual group in the sense of Voiculescu. This structure allows to define the free and the tensor convolution of two states. In a joint work with Michael Ulrich, we proved that there exist no Haar state for those two convolutions. However, it is possible to define a weaker notion of Haar state by restricting us to tracial states: there exists a Haar trace for each convolution, in the sense that it is an absorbing element in the set of tracial states. I will describe those Haar traces and enlighten some links between the unitary dual group and the random matrices.
- 20.05.2015, Antoine Dahlqvist (Berlin)
Large N-asymptotics of a matrix-valued diffusion: the unitary Brownian motion
A unitary Brownian motion is a continuous process on a finite dimensional unitary group whose multiplicative increments are stationary and independant. We shall focus here on the ones that are invariant by adjunction as the dimension goes to infinity. When these processes are properly scaled, their non-commutative distribution converges to the one of a free Brownian motion. We shall explain here that the latter convergence is strong and that its fluctuations are gaussian. As a corollary of this result, the unitary brownian motion has a hard-edge. In other words, the spectrum of a marginal of a unitary brownian motion converges in Haussdorff distance to the one of the same marginal of the free Brownian. The strong convergence result is part of a joint work with Benoît Collins and Todd Kemp.
- 27.5.2015, Jamie Mingo (Queen's University, Kanada)
Wigner Matrices and the Graphical Calculus
Roland Speicher and I found an efficient method of
analyzing some sums which regularly appear in random
matrix theory. The method reduces a crucial calculation to a
simple one involving graphs. I will explain our method and
how it can be used to obtain asymptotic results for some
deformed random matrix ensembles.
- 3.6.2015, Camille Male (Paris, Frankreich)
Moment method for adjacency matrices of large random graphs (joint work with Prof. S. Peche)
We study the asymptotic freeness properties of adjacency matrices of large graphs with dependent entries when the number of vertices N goes to infinity. In particular, we focus on the model of uniform regular simple graph G_N with large degree d_N, going to infinity with N. The difficulty of this model is that we do not have formulas for the joint distribution of entries of such random matrices.
We give a new proof of the convergence of the spectral distribution of G_N toward the semicircular law (a result known by Tran, Vu, Wang, RSA 2013) and prove the asymptotic freeness of its adjacency matrix with a large class of deterministic matrices.
We use the moment method à la Wigner. We prove a general criterion which quantify the fact that a graph with "decorrelated edges" converges to a semicircular variables and is asymptotically free with deterministic matrices. We prove that this criterion holds for the uniform regular graph using a symmetry of the model which is referred as the switching method.
- 3.6.2015, Franck Gabriel (Paris, Frankreich)
The partition algebra, cumulants and convergence of Levy processes
The notion of cumulants is a very convenient tool in probability and free probability. Using a special geometry on the algebra of partitions $\mathcal{P}_{2k}$ on $2k$ elements, these two notions can be unified. Besides, this set of partitions also allows to define two other notions of cumulants: one suited for sequences of matrices which are conjugation invariant by orthogonal matrices, the second one for these which are conjugation invariant by permutation matrices.
We will talk about the consequences of these new notions, explaining how they allow to recover very quickly some well-known theorems in the large random matrices theory. We will also explain a general theorem for the convergence in probability of observables of additive and multiplicative matricial Lévy processes which are conjugation-invariant by the symmetric group. Depending on the time remaining, we will show in this setting the convergence of the simple random walk on the symmetric group.
- 24.6.2015, Jurij Volcic (Auckland, Neuseeland)
Matrix coefficient realization theory of noncommutative rational functions
One of the main reasons why noncommutative rational functions, i.e.,
elements of the universal skew field of fractions of a free algebra, are so
inaccessible, is their lack of a canonical form. For rational functions defined
at 0 this can be compensated by using realizations, which originated in automata
theory and systems theory. The aim of this talk is to present a realization
theory that is applicable to every noncommutative rational function and is
adapted for studying its finite-dimensional evaluations. Using these matrix
coefficient realizations we can measure the complexity of noncommutative
rational functions, describe their singularities and assert size bounds for the
rational identity testing problem. The self-adjoint version will be also considered.
- 8.7.2015, Sheng Yin (Besancon, Frankreich)
Mazur maps for noncommutative L_p spaces
It is well known that the Mazur map from L_p to L_q is (p/q)-Hölder if 1≤ p< q<∞ and is Lipschitz on the unit ball if 1≤ q< p<∞. So we expect a similar behavior for the Mazur maps for noncommutative L_p spaces. And this result is true in full generality, given by a recent paper of Ricard. In this talk, it will follow the techniques developed by Ricard to show the result for Schatten classes.
- 29.7.2015, Pierre Tarrago (Saarbrücken)
Free wreath product and spin planar algebras
In this talk I will present a link between the free wreath product of non-commutative permutation groups and a particular operation on planar algebras. Using this equivalence I will show that the character law of a free wreath product is the free multiplicative convolution of the initial character laws.
This is a joint work with Jonas Wahl.
- 29.7.2015, Carlos Vargas Obieta (Graz, Österreich)
Block-modified random matrices and OVFP.
From an md by md unitarily invariant random matrix ensemble and a fixed
linear map \phi, from the spaces of m by m matrices to the space of n by n matrices, we construct the nd by nd block-modified random matrix X^{\phi}=(\phi \otimes id_d)(X).
In order to compute the asymptotic distribution (d\to\infty) of the modified ensemble, one needs to understand the joint asymptotic distribution of the d by d blocks (x_ij), i,j\leq m of X. For example, a single diagonal block has the distribution of the free compression of X, and the joint moments of the blocks are computed similarly.
Once this joint distribution is understood, the block modification is simply an expression of the form \sum b_ij\otimes x_ij. We give several examples of maps where the asymptotic distribution of X^{\phi} can be computed.
- 4.8.2015, Tomohiro Hayase (Tokyo, Japan)
De Finetti theorems for a Boolean analogue of easy quantum groups
We define a new kind of category of partitions and associated
Boolean analogue of easy quantum groups. Then we prove de Finetti type
theorems for them which imply conditional Boolean independence and other
distributional restrictions. .
