Prof. Dr. Roland Speicher
Dr. Octavio Arizmendi Echegaray
Oberseminar zur Freien Wahrscheinlichkeit
In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeit.Zeit und Ort
mittwochs, 16-18 Uhr, SR10 (316)Vorträge
- Dienstag (!), 9.4.2013, 14h ct, SR6 (217) Gaetan Borot, MPIM Bonn
Analytic combinatorics of maps of all topologies
Maps are discrete surfaces obtained by gluing polygons along their edges. The enumeration of rooted planar maps has first been addressed by Tutte in the 60s, thanks to a recursive decomposition of maps which translates into a quadratic functional equation for the generating series of rooted planar maps, which can be solved explicitly. By similar methods, one can derive other (again quadratic) functional equations for the generating series of maps of higher topologies, which are solved by a universal "topological recursion" formula. The initial data of the recursion consists of the generating series of disks (genus 0, 1 rooted boundary) and of cylinders (genus 0, 2 rooted boundaries). Universal here means that the same "topological recursion" also enumerate maps carrying certain statistical physics model (like the Ising model, or the O(n) loop model), provided the initial data is properly chosen. In this talk, I will present the solution of the enumeration problem by the topological recursion for a class of O(n) loop models. This is partly based on joint work with B. Eynard and N. Orantin.
By definition of the model above, it is allowed to identify two edges of the rooted boundary. Then, it is possible to deduce the generating series of maps with simple boundaries, i.e. for which two edges of the same boundaries cannot be identified. At the planar level, the relation between the two generating series is the R-transform. This relation can be explicitly described for all topologies, and it enjoys algebraic properties which still need to be elucidated. In the second part of the talk, I will present these (yet unpublished) observations.
- 10.4.2013, Claus Koestler
The Thompson group F from the viewpoint of noncommutative probability
We characterize the extremal characters of the Thompson group F. Our approach is inspired from techniques in the context of distributional symmetries in noncommutative probability. In particular we address noncommutative independence in the left regular representation of the Thompson group F. This is joint work with Rolf Gohm.
- 29.5.2013, Takahiro Hasebe
Free infinite divisibility for beta distributions
Many beta distribution of first and second kind are shownto be feely infinitely divisible. This generalizes the result of Arizmendi and Belinschi. Taking limits, we can show that many gamma and inverse gamma distributions are freely infnitely divisible too.
- 19.6.2013, Rosaria Simone
Universality results for polynomials in freely independent random variables
The aim of the talk is to provide an overview of the universality results available for polynomials in freely independent random variables, with comparison with the classical probability counterparts. It is known that homogeneous sums based on a sequence of freely independent semicircular elements behave universally with respect to semicircular approximation. Thanks to a general invariance principle and to the celebrated Fourth Moment Theorem, other examples of universal laws are given, with respect to both semicircular and free Poisson approximation, and in both a unidimensional and multidimensional setting. Finally, new persepectives about a possible characterization of universal laws, based on a fourth moment condition, are presented.
- 26.6.2013, Cedric Schonard
Free monotone transport
In 2012 A. Guionnet and D. Shlyakhtenko have shown that there exists a non-commutative analog of Brenier's monotone transport theorem. We can view such a free monotone transportation map as certain kind of replacement of density of a non-commutative law. In this talk I want to present the main ideas of their paper and I will discuss some applications.
Aktualisiert am: 25.6.2013 Octavio Arizmendi