Research seminar Free Probability Theory
In this research seminar we treat topics ranging from free probability and random matrix theory to combinatorics, operator algebras, functional analysis and quantum groups. |
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Talks in 2015
- 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
- [1] Daniel Alpay, Palle Jorgensen, Guy Salomon:
- 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.
- 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 ŕ 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. .
- 21.10.2015, SR 6 Ping Zhong (Lancaster).
Quasi-free Quantum Stochastic Calculus and Quantum Random Walks
Attal and Joye considered the repeated interaction model of quantum random walks and studied the continuous limit for repeated quantum interactions with a sequence of identical quantum system in a given faithful normal state. They pointed out that thermal quantum noises appearing in the limit can be understood as quantum stochastic integrals against fundamental (time, creation, annihilation) processes obtained by non-Fock representation of CCR algebras. The theory of non-Fock quantum Brownian motion and corresponding quantum stochastic calculus, which generalized Hudson-Parthasarathy's quantum stochastic calculus, was developed by Hudson and Lindsay for Gauge-invariant quasi-free states on CCR algebras and was extended recently further by Lindsay and Margetts to include general states on the CCR algebras. Joint with Belton, we show that certain thermal noises including Attal and Joye's example can be squeezed by Bougoliubov transformations on the CCR algebra, and we address the uniqueness question of quasi-free states imposed on the CCR algebra. In other words, we can only detect temperature and we can not recognize whether the thermal noises have been squeezed. We obtain quasi-free dynamics for a class of examples which extend that of Attal and Joye.
- 5.11.2015 (Donnerstag 10 Uhr c.t.), SR 6 Miguel Ángel Pluma Rodríguez (CIMAT, Mexiko).
Ulam's Problem, Brownian Functionals and the Spectrum of Gaussian Unitary Matrices
In 1961 Ulam proposed the following problem: pick a random permutation of the numbers $1,...,n$ with uniform distribution and consider the length of the longest increasing subsequence $LI_n$. The problem to find the asymptotic distribution of $LI_n$ is known as the Ulam's problem. In 1999 by Baik, Deift and Johansson gave the solution to Ulam's problem in terms of Tracy-Widom distribution. Since then, a different variant to Ulam's problem have been studied,one of them the so called Ulam's problem for random words. The objective of this talk, is present the required tools and the main ideas in the solution to Ulam's problem for random words in terms of the maximum eigenvalue of a Gaussian Unitary matrix, and as a Brownian functional. As a consequence, we will obtain a representation of the maximum eigenvalue of a Gaussian Unitary matrix as a Brownian functional. Finally we present some generalizations to Ulam's problem, and a conjecture about the representation of the maximum eigenvalue of a Gaussian Unitary matrix with a covariance structure in the diagonal entries as a Brownian functional.
- 25.11.2015 (14 Uhr c.t.), SR 7 John Williams (Saarbruecken) .
Hausdorff Continuity of Free Convolution Semigroups
Let μ denote a Borel probability measure and let {μt}t≥1 denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for t>1. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang. Extensions to multiplicative convolution will be discussed.
- 2.12.2015 , SR 6 Pierre Tarrago (Tours, Frankreich) .
Free product of planar algebras and the Belinschi-Nica semigroup
In the first part of this talk, I will recall some basic results on Jones' planar algebras: in particular I will show that each planar algebra is fully described by its associated graph, the so-called principal graph.
In a second part, I will use the concept of principal graph and a work from Belinschi and Nica to compute the Poincaré serie of a free product of planar algebras.
- 9.12.2015 , SR 6 Camille Male (Paris, Frankreich) .
Freeness and the Fourier transform
Motivated by question from telecommunication, we study the relation between a diagonal matrix A_N with independent entries and a matrix B_N of the form B_N= U_N D_N U_N^*, where D_N is a diagonal matrix independent of A_N and U_N is the so called Fast Fourier Transform matrix
U_N = 1/sqrt N ( omega^(i-1)(j-1))_{i,j=1..N} where omega is a primitive N-th root of the unity.
It is known from Tulino, Caire, Shamai, Verdu and from Farell and Anderson that if D_N has independent entries then A_N and B_N are asymptotically free. We will see that this is no longer true when the entries of D_N as sufficiently correlated, and use the notion of asymptotic traffic independence to describe and compute the limiting joint distribution.
- 16.12.2015 , SR 6 Wiktor Ejsmont (Wroclaw, Poland) .
Sample variance in free probability
A new characterizing property of the sample variance if free probability is proved whichcan, in essence, be summarized as follows. Let $\X_1, \X_2,\dots, \X_n$denote free independently and identically distributed random variables with finite non-zero variance $\sigma^2$. Then $\sum_{i=1}^n(\X_i-\overline{\X})^2/\sigma^2$is distributed as free chi-square distribution with $n -1$degrees of freedom, for some fixed $n\geq 2$if and only if $\X_1,\dots,\X_n$are odd.
It is worth mentioning that this problem come from classical probability and it has been posed as an unsolved by Kagan, Linnik, and Rao(this characterization problem is still open in classical probability).
Aktualisiert am: 6. November 2017 Moritz Weber