### Project 01: "Random walk models on graphs and groups"

*This project is running since 2010.*

Principal investigator: Wolfgang Woess

Graz University of Technology, Austria.

*Mentor for*: Koch; Bazarova, Ćustić, Ebner.

Associated scientist: Franz Lehner

Graz University of Technology, Austria.

*Mentor for*: Boiko, Candellero, Carl, Cuno.

Associated scientist: Wilfried Imrich (since 2015)

University of Leoben, Austria.

*Mentor for*: Cuno, Lehner.

### DK Student

**Second phase of the doctoral program:****Gundelinde Wiegel**(Germany; since May 2016)

*Mentors*: Christian Elsholtz, Ecaterina Sava-Huss.**First phase of the doctoral program:****Johannes Cuno**(Germany; May 2011–October 2015)

Personal homepage;*Email*: cuno@math.tugraz.at

*Mentors*: Wilfried Imrich, Bettina Klinz, Franz Lehner.

*PhD Defense*: October 14, 2015.

*Referees*: T. Riley (Cornell), V. Kaimanovich (Ottawa), W. Woess.

*Examiners*: T. Riley (Cornell), V. Kaimanovich (Ottawa).

### Associated Students

**Second phase of the doctoral program:****Judith Kloas**(Germany; since March 2014)

*Email*: kloas@math.tugraz.at

*Mentors*: Mihyun Kang, Sebastian Müller.**First phase of the doctoral program:****Tetiana Boiko**(Ukraine; October 2010–October 2014)

*Email*: boiko@math.tugraz.at

*Mentors*: Franz Lehner, Johannes Wallner.

*PhD Defense*: October 17, 2014.

*Referees*: A. Bendikov (Wroclaw), M. Salvatori (Milano), W. Woess.

*Examiners*: A. Bendikov (Wroclaw), M. Salvatori (Milano).**Elisabetta Candellero**(Italy; July 2010–July 2012)

Personal homepage;*Email*: candellero@math.tugraz.at

*Mentors*: Istvan Berkes, Franz Lehner.

*PhD Defense*: July 9, 2012.

*Referees*: S. Lalley (Chicago), N. Gantert (Munich), W. Woess.

*Examiners*: S. Lalley (Chicago), W. Woess.**Christoph Temmel**(Austria; July 2010–March 2012)

Personal homepage;*Email*: math@temmel.me

*Mentors*: Istvan Berkes, Pierre Mathieu (Marseille).

*PhD Defense*: March 19, 2012.

*Referees*: J. v. d. Berg (Amsterdam), P. Mathieu (Marseille), W. Woess.

*Examiners*: J. v. d. Berg (Amsterdam), P. Mathieu (Marseille).

### Project description (pdf-file)

The central topic of the research of W. Woess is Random Walks on Infinite Graphs and Groups. Here, random walks are understood as Markov chains whose transition probabilities are adapted to an algebraic, geometric, resp. combinatorial structure of the underlying state space. The main theme is the interplay between probabilistic, analytic and potential theoretic properties of those random processes and the structural properties of that state space. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk, such as transience/recurrence, decay and asymptotic behaviour of transition probabilities, rate of escape, convergence to a boundary at infinity and harmonic functions. Vice versa, random walks may also be seen as a nice tool for classifying, or at least describing the structure of graphs, groups and related objects. These competences are complemented by the expertise in Non-commutative Probability and Functional Analysis of the associated scientist F. Lehner.