## Prof. Dr. Roland Speicher

## Prof. Dr. Moritz Weber

### Dr. Guillaume Cebron

## Oberseminar zur Freien Wahrscheinlichkeit

In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeit.### Zeit und Ort

Mittwochs, 14-16 Uhr, SR7*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.

**(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.

**(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.

*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.

*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.

*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).

*Representation theory of subfactors and quantum groups*

Every finite index inclusion of II_1-factors gives rise to a powerful combinatorial structure called the standard invariant. This invariant has been abstractly characterized in several different ways, including Popa's lambda-lattices and Jones' planar algebras. In this talk, I will present recent results of Popa and Vaes on the representation theory of the standard invariant. As an application, I will show how to combine these results with the planar algebra formalism to deduce approximation properties of certain quantum permutation groups.

*On a model for boolean-free independence*

We shall describe a construction of (algebras of) operators on a Hilbert space, which are index by a partially ordered set, and which satisfy the following properties: 1. if the index set is totally ordered, then the algebras are freely independent 2. if the index set is totally disordered, then the algebras are boolean independent. This construction is intended to develop an abstract notion of bf-independence, which would combine the boolean and the free ones. Moreover, for the partial order on the $d$-dimensional euclidian space $R^d$, given by the positive cone $R^d_+$, we shall present the associated Central Limit Theorem for random variables indexed by the partially ordered lattice $N^d_+$ in $R^d$.

*The Two-Parameter Free Unitary Segal-Bargmann Transform*

We derive an integral kernel of the two-parameter free unitary Segal-Bargmann transform which takes the $L^2$ space of the law of the free unitary Brownian motion on the unit circle to a reproducing kernel Hilbert space of holomorphic functions on a region on the complex plane. The integral transform coincides with the large-$N$ limit of the two-parameter Segal-Bargmann-Hall transform on $\U(N)$. We introduce the two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin identification. We also give a conditional expectation representation of the transform. This work combines, extends, and completes the story started by Biane and continued by Cébron and Driver-Hall-Kemp.

*The Norm of Non-commutative Polynomials in Independent Gaussian Random Matrices*

We know that the Gaussian random matrices form a good model for freely independent semicircular elements in a C*-probability space in the view of moments. In Haagerup and Thorbjørnsen's paper, A new application of random matrices: Ext(C*_red(F_2)) is not a group (2005), they proved that for any non-commutive polynomial in independent Gaussian random matrices, its norm convergences almost surely to the norm of this polynomial evaluated at freely independent semicircular elements. In the paper, they use the technique like linearization trick and comparison of Cauchy transforms, which will be highligted in this talk.

**(10 Uhr c.t.), SR 9**Sutanu Roy (Ottawa, Canada) .

*Braided quantum SU(2) groups*

We construct a family of q-deformations of SU(2) for non-zero complex deformation parameter q. For real q, deformation coincides with compact quantum group SUq(2) group introduced by S.L. Woronowicz. For non real q, Suq(2) is a braided compact quantum group. This a joint work with Pawel Kasprzak, Ralf Meyer and Stanislaw Lechs Woronowicz.

**(14 Uhr s.t.), SR 6**Raj Rao Nadakuditi (Michigan, US) .

*New applications of random matrix theory*

We describe some new success stories where random matrix theory and free probability theory has enabled new applications: these include new theory and algorithms for transmitting light perfectly through highly scattering media and for separating foreground and background of videos in highly cluttered scenes. We conclude by highlighting some newly discovered random matrix universality phenomena emerging from scattering theory and semidefinite optimization and connections to free probability.

**(15:30 Uhr), SR 6**Amaury Freslon (Paris-Sud, Frankreich) .

*Maximal abelian subalgebras in free (quantum) groups*

I will explain how one can investigate the structure of the so-called laplacian subalgebra in free quantum group factors based on ideas from the case of free group factors. This enables to prove that this algebra is maximal abelian and singular as well as to compute the associated measure class.

Aktualisiert am: 29. Februar 2016 Guillaume Cebron | Impressum |