### 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.

*PhD Defense*: March 2, 2018.

*Referees*: W. Imrich (Leoben), Marc Peigné (Tours), W. Woess.

*Examiners*: W. Imrich (Leoben), Marc Peigné (Tours).**Christian Lindorfer**(Austria; since April 27, 2018)

*Email*: c.lindorfer@student.tugraz.at

*Mentors*: Karin Baur, Birgit Vogtenhuber.**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", which is also the title of the quite successful monograph [2000; paperback re-edition from 2008]. 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. More recent interests also comprise random processes on complexes that arise from graphs, such as the strip complexes of the three papers by Bendikov, Saloff-Coste, Salvatori and Woess [2011, 2015, 2016], as well as processes on ultrametric spaces, viewed as boundaries of trees, see Bendikov, Girgoryan, Pittet and Woess [2014] and Bendikov, Cygan and Woess [2017] . There, a particularly attractive issue is the duality between those processes and random walks on the corresponding trees. Further recent work concerns, e.g., random walks in buildings - a collaboration with J. Parkinson [2015] - and random walk-related issues from potential theory on trees and the hyperbolic plane, see Boiko and Woess [2015] and Picardello and Woess [2017] . The work of W. Woess is not limited to the interplay between random processes and structure theory. For example, there is also a body of more "pure" work on infinite graphs, group actions, and in particular formal languages (which entered the scene via the free group). See for example Ceccherini-Silberstein and Woess [2012] and Woess [2012] . Woess' research is interdisciplinary between several mathematical areas: Probability - Graph Theory - Geometric Group Theory - Discrete Geometry - Potential Theory - Harmonic Analysis and Spectral Theory.