Research seminar Free Probability Theory

Oberseminar zur Freien Wahrscheinlichkeitstheorie

 In this research seminar we treat topics ranging from free probability and random matrix theory to combinatorics, operator algebras, functional analysis and quantum groups.

Time and place

Wednesdays, 14:15-16:00, room SR 6, building E2 4
talks are 60 minutes plus discussion

Talks in 2017

 November 22, 2017, 2:15 pm, Johannes Alt (Vienna) Local inhomogeneous circular law The density of eigenvalues of large random matrices typically converges to a deterministic limit as the dimension of the matrix tends to inﬁnity. In the Hermitian case, the best known examples are the Wigner semicircle law for Wigner ensembles and the Marchenko-Pastur law for sample covariance matrices. In the non-Hermitian case, the most prominent result is Girko’s circular law: The eigenvalue distribution of a matrix X with centered, independent entries converges to a limiting density supported on a disk. Although inhomogeneous in general, the density is uniform for identical variances. In this special case, the local circular law by Bourgade et al. shows this convergence even locally on scales slightly above the typical eigenvalue spacing. In the general case, the density is obtained via solving a system of deterministic equations. In my talk, I explain how a detailed stability analysis of these equations yields the local inhomogeneous circular law in the bulk spectrum for a general variance proﬁle of the entries of X. This result was obtained in joint work with László Erdős and Torben Krüger. 8.11.2017, 14:00 st, Julien Sazadaly (Reims) Quantum isometries of partially commutative spheres Considering the framework of C*-algebras, one can consider the generalization of spheres. Those one, introduced by Roland Speicher and Moritz Weber, may be endowed, first, with a structure of non-commutative algebraic manifold. As a result, one can compute the linear automorphism quantum group, acting by matrix comultiplication. In the case of partially commutative sphere, this group has been computed by Roland and Moritz (partially commutative orthogonal group). On the other hand, with a suitable definition which is a slight rewriting of Goswami and Bhowmick one’s (quantum group of isometries), we will endow those spheres with a « Laplacian-type » spectral triple, miming the classical Riemaniann structure. By using the toolbox grounded by Goswami and Bhowmick, it will also be possible to construct the notion of quantum isometry group. Then, we will show, by proving a link between the precedent « algebraic » result and the machinerie of quantum isometry group, that the linear automorphism quantum groups of the partially commutative spheres (partially commutative quantum orthogonal group) and their quantum isometry group are equal (this is a retailoring and a generalization of an earlier result, proved by Goswami and Banica). 8.11.2017, 15:00 st, Simeng Wang (Saarbrücken) Unitary quantum groups and noncommutative complexe spheres with partial commutation relations In a recent paper by Speicher and Weber, the noncommutative real spheres with mixed commutation relations are studied, and the associated quantum automorphism groups are computed. In this talk we will generalize these notions to the complex setting, and compute the quantum symmetries on the complex epsilon-spheres. 18.10.2017,  Adrian Celestino    (CIMAT, Mexico) Zeit: 14:00 Uhr c.t.   Ort: Seminarraum 6, Geb. E2 4 Cumulants in Free Probability of type B Free Probability of Type B was introduced by Biane, Goodman, and Nica (2003) in order to create an analogous to Free Probability Theory, where the lattices of non-crossing partitions are replaced by the lattices of non-crossing partitions of type B. In this talk, we will see how this was done by the authors. More specifically, we will see how they use boxed convolution as a bridge between Cayley graphs of Symmetric groups and Free Probability Theory. Then, replacing Symmetric groups by Hyperoctahedral groups, they define cumulants of type B and a notion of free independence which is characterized by vanishing mixed cumulants of type B 11.10.2017,  Mauricio Salazar    (CIMAT, Mexico) Zeit: 15:00 Uhr s.t.   Ort: Seminarraum 6, Geb. E2 4 On the Rate of convergence for the Boolean and Monotone CLT The Classical Berry-Esseen Theorem states that the rate of convergence in the Central Limit Theorem, on the Kolmogorov metric, is of order 1/sqrt(n). An analogous result for the free CLT was obtained by Kargin in 2007 for measures of bounded support, being the rate of convergence also 1/sqrt(n). In 2008 Chystiakov and Götze improved that result only requiring that the fourth moment exists. In this talk I will speak about some results on the rate of convergence for the Boolean and monotone Central Limit Theorems. 11.10.2017,  Tulio Gaxiola    (CIMAT, Mexico) Zeit: 14:00 Uhr s.t.   Ort: Seminarraum 6, Geb. E2 4 Spectral Analysis of Growing Graphs and Quantum Probability We talk about spectral analysis of large graph (or of a growing graph). We show how different products of graph are related to notions of independence. We also describe asymptotic distribution of distance-k graph of products. Finally we show how quantum probabilistic techniques are applied for the study of asymptotics of spectral distributions 14.06.2017,  Amaury Freslon   (Orsay, Frankreich) Zeit: 14:00 Uhr c.t.   Ort: Seminarraum 6, Geb. E2 4 The classification of two-coloured partition quantum groups Building on ideas of T. Banica and R. Speicher, one can associate to a suitable collection of partitions with colours a compact quantum group. This is a rich source of examples which covers most of the quantum groups studied so far. It is therefore an interesting problem to classify the objects arising from this construction. Up to now, the strategy to prove classification results is to use combinatorial arguments to list the possible sets of partitions, independently of the associated quantum group. In this talk I will explain another strategy relying on the representation theory of the quantum groups and replacing most combinatorial arguments by algebraic computations in fusion rings. As an application, I will give a classification of all partition quantum groups on two slef-inverse colours. Together with the works of M. Weber, S. Raum and P. Tarrago this completes the classification of all partition quantum groups on two colours. 31.05.2017,  Raj Rao    (Michigan, USA) Zeit: 16:00 Uhr c.t.   Ort: Seminarraum 10, Geb. E2 4 Free component analysis We describe a method for unmixing mixtures of 'freely' independent random variables in a manner analogous to the indepedent component analysis (ICA) based method for unmixing independent random variables from their additive mixture. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from an additive mixture of matrices. We describe the theory -- the various 'contrast functions', computational methods and compare FCA to ICA on data derived from real-world experiments. 31.05.2017,  Jonas Wahl    (Leuven, Belgium) Zeit: 15:00 Uhr s.t.   Ort: Seminarraum 6, Geb. E2 4 Non-singular Bernoulli actions and L^2-cohomology The study of probability preserving Bernoulli actions of discrete groups has a rich history in ergodic theory, dating back to von Neumann and Kolmogorov. However, as soon as one steps away from the probability preserving case, the results on such actions are scarce in spite of their prototypical nature. In fact, examples of Bernoulli actions of type III have only be given recently by Kosloff and Danilenko/Lemanczyk and only for the group of integers. I will present a joint article with Stefaan Vaes wherein we prove the following conjecture for almost all infinite discrete groups: An infinite discrete group admits a Bernoulli action of type III if and only if its first L^2-cohomology is non-zero. 20.03.2017,  Mario Alberto Diaz Torres    (Queen's University, Kingston) Zeit: 16:00 Uhr c.t.   Ort: Seminarraum 6, Geb. E2 4 Some remarks on the facets of second-order free probability Extending a phrase by J. Mingo and R. Speicher [1], free probability has at least four basic facets: operator algebras, free harmonic analysis, random matrix theory, and combinatorics. In this talk we will discuss some work in progress with Jamie Mingo and Serban Belinschi concerning the random matrix theory and combinatoric facets of second-order free probability. As we will see, this approach may shed some light on problems concerning the other two facets. [1] J. Mingo and R. Speicher. "Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces." Journal of Functional Analysis, 2006. 20.03.2017,   Jamal Najim    (Marne-la-Vallee, Frankreich) Zeit: 14:00 Uhr c.t.   Ort: Seminarraum 6, Geb. E2 4 Non-hermitian random matrices with a variance profile For each n, let $A_n = (\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n = (X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We are interested in the asymptotic behavior of the empirical spectral distribution μYn of the rescaled entry-wise product $$Y_n = \frac 1{\sqrt{n}}σ_{ij}X_{ij} .$$ For our main result, we provide a deterministic sequence of probability measures μn, each described by a family of Master Equations, such that the difference μYn −μn converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries σij to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property. This is a joint work with Nick Cook, Walid Hachem and David Renfrew. 14.03.2017,   Jurij Volcic    (University of Auckland, Neuseeland) Zeit: 14:00 Uhr c.t.   Ort: Seminarraum 7, Geb. E2 4 Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras A commutative ring embeds into a field if and only if it has no zero divisors; moreover, it this case it admits a unique field of fractions. On the other hand, the problem of localization of noncommutative rings and embeddings into skew fields (that is, division rings) is much more complex. For example, there exist noncommutative rings without zero divisors that do not admit embeddings into a skew field, and rings with several non-isomorphic "skew fields of fractions". This lead Paul Moritz Cohn to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of free algebras. With the help of matrix evaluations we will construct the skew field of multipartite rational functions, which turns out to be the desired universal skew field of fractions. We will also explain its role in the difference-differential calculus in free analysis. 14.03.2017,   Ken Dykema    (Texas A&M University, USA) Zeit: 11:00 Uhr c.t.   Ort: Seminarraum 6, Geb. E2 4 Commuting operators in finite von Neumann algebras We find a joint spectral distribution measure for families of commuting elements of a finite von Neumann algebra. This generalizes the Brown measure for single operators. Furthermore, we find a lattice (based on Borel sets) consisting of hyperinvariant projections that decompose the spectral distribution measure. This leads to simultaneous upper triangularization results for commuting operators. (Joint work with Ian Charlesworth, Fedor Sukochev and Dmitriy Zanin.) 22.02.2017,   Konrad Schrempf    (TU Graz, Österreich) Linearizing the Word Problem in (some) Free Fields We are interested in constructing minimal linear representations of elements in the universal field of fractions (free field) of the free associative algebra (over a commutative field). I will present a recent result which serves as a first step, namely solving the word problem with linear techniques and discuss some open problems. Since the factorization of non-commutative polynomials seems to be closely related, I try to sketch a possible setup. 08.02.2017,   Mario Kieburg    (Universität Bielefeld, Deutschland) Zeit: 10:30 Uhr s.t.   Ort: Seminarraum 8, Geb. E2 4 Products of Polynomial Ensembles In the past years a revival of products of random matrices has happened. This is due to the exact calculation of their joint probability densities of the eigenvalues as well as the singular values at finite matrix dimension and at a finite number of matrices multiplied. One question remained open why these exact calculations were possible. Kuijlaars et al. quite early identified a particular structure of the joint probability density functions which all these example of products as well as sums of random matrices satisfy. This class of ensembles is called polynomial ensembles. In a work of last year by Holger Koesters and me, we showed that the joint probability densities of the eigenvalues and the one of the singular values of a bi-unitarily invariant random matrix ensemble (for example the Ginibre ensemble) are bijectively related. As a byproduct of this result we could identify a proper subset of the polynomial ensembles. This set has the nice property that it builds a semi-group with respect to the multiplication of the corresponding bi-unitarily invariant random matrices and even has a semi-group action on the whole set of polynomial ensembles. I am going to talk about the consequences of this observation and a first but failed approach to analytically continue the number of matrices multiplied. I will also point out a way to circumvent the problems arising in such an analytical continuation. 01.02.2017,   Sheng Yin    (Universität des Saarlandes, Deutschland) Rational function in strongly convergent random variables In this talk, we want to show that for a strongly convergent sequence of random variables, the convergence of trace and norm can be extended from polynomials to rational functions under certain assumptions. Roughly speaking, it claims that the strong convergence property is stable under taking the inverse of non-commutative random variables which have this property. As a direct corollary, we can conclude that for any rational function which is well-defined at a tuple of freely independent semi-circular elements, we have the convergence of trace and norm of this rational function in independent GUE random matrices to its evaluation at these semi-circular elements, almost surely. Such a result is expected in a recent paper by Helton, Mai and Speicher, where they can calculate the distribution of a rational expression in non-commutating random variables and their simulation indicates the convergence of corresponding random matrices should be true. 04.01.2017,   Matthieu Josuat-Verges    (Marne-la-Valle, Frankreich) Free cumulants and Schöder trees It is now well established that free cumulants, defined in terms of noncrossing partitions, give a combinatorial point of view on free probability. In the case of free cumulants of a single random variable, we can alternatively use the R-transform, an operation on functions that is essentially the compositional inversion. In this work we give an extension of the R-transform that covers the multi-variable case, using ideas coming the theory of operads, and an appropriate multilinear extension of free cumulants. We get in particular new combinatorial formulas involving Schröder trees rather than noncrossing partitions. (Joint work with Frédéric Menous, Jean-Christophe Novelli, Jean-Yves Thibon). 04.01.2017,   Guillaume Cebron    (Touluse, Frankreich) Stein kernels and the fourth moment theorem We will review some functional inequalities which are useful in order to measure the closedness of a random variable to the semicircular variable. It will allow us to give a quantitative version of the fourth moment theorem of Kemp-Nourdin-Peccati-Speicher. This is joint work with Tobias Mai.

Past talks

2016
2015
2014
2013
2012
2011

 Aktualisiert am: 14. November 2017   Tobias Mai Impressum