### Talks in 2012

#### Vortrag im Seminar Diskrete Mathematik und Optimierung

**Title:**Random Walks on critical Percolation-Trees

**Speaker:**Florian Sobieczky (University of Denver)

**Date:**18.12. 2012, 14:15

**Room:**Seminarraum C208, Steyrergasse 30, 2. Stock

**Abstract:**

Bounds for the expected return probability of the delayed random walk on

finite clusters of an invariant percolation on transitive unimodular

graphs are derived. They are particularly suited for the case of critical

Bernoulli percolation and the associated heavy-tailed cluster size

distributions. The upper bound relies on the fact that cartesian products

of finite graphs with cycles of a certain minimal size are Hamiltonian.

For critical Bernoulli bond percolation on the homogeneous tree this bound

is sharp. The asymptotic type of the expected return probability for large

times t in this case is of order of the 3/4'th power of 1/t.

#### Joint Seminar

**Title:**Lumped Markov chains and entropy rate preservation

**Speaker:**Bernhard Geiger and Christoph Temmel (TU Graz)

**Date:**10 Dez, 16:00

**Room:**IDEG134

**Abstract:**

A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterize the entropy rate loss of a lumping of a stationary Markov chain on a finite state space in two ways. First, by the asymptotic ratio of the number of trajectories with positive weight between the original and the lumped chain. Second, by the reconstructability of original trajectories from their images under the lumping. Every non-trivial lumping of a Markov chain with positive transition matrix incurs an entropy rate loss. We give sufficient conditions on the non-positive transition matrix and the lumping to preserve the entropy rate. In the sparse setting we state sufficient conditions on the lumping to both preserve the entropy rate and result in a k-th order homogeneous Markov chain.

#### Kolloquium: Mathematische Methoden in den Natur- und Ingenieurwissenschaften

**Title:**Gauss, Jacobi, Seidel, Richardson, Krylov: The Invention of Iterative Methods

**Speaker:**Univ.-Prof. Dr. Martin J. GANDER (Universität Genf)

**Date:**Freitag, 7.12.2012, 16:00 Uhr

**Room:**TU Graz, Steyrergasse 30, 3. Stock, Seminarraum C307

**Abstract:**

#### Vortragseinladung

**Title:**Optimal Transport, Model-Indepedence, and Trajectorial Inequalities.

**Speaker:**Mathias Beiglböck (Universität Wien)

**Date:**Freitag, 7.12.2012, 14:15 Uhr

**Room:**Seminarraum A206, Steyrergasse 30, 2.Stock, Geodäsie

**Abstract:**

Abstract:

We will explain a recently discovered connection between Optimal

Transport and the areas of model independence / martingale

inequalities in probability. This link has a number of fruitful

consequences. For instance, the duality theorem from optimal transport

leads to new super-replication results. Optimality criteria from the

theory of mass transport can be translated to the martingale setup and

allow to characterize minimizing/maximizing models in finance.

Moreover, the dual viewpoint provides new

insights to the classical inequalities of Doob and Burkholder-Davis-Gundy.