### 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*: Minelli, Lipnik; 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 Students

**Third phase of the doctoral program:****Panagiotis Spanos**(Greece; since September 2019)

*Email*: spanos@math.tugraz.at

*Mentors*: Peter Grabner, Christoph Aistleitner.

**Second phase of the doctoral program:****Gundelinde Wiegel**(Germany; May 2016–September 2020)

*Mentors*: Christian Elsholtz, Sebastian Müller.

*Thesis Title*: "Random processes on trees".

*PhD Defense*: September 14, 2020.

*Referees*: N. Gantert (München), W. Imrich (Leoben), W. Woess.

*Examiners*: W. Imrich (Leoben), W. Woess.**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.

*Thesis Title*: "Random walks of Baumslag-Solitar groups".

*PhD Defense*: October 14, 2015.

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

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

### Associated Students

**Third phase of the doctoral program:****Stefan Hammer**(Austria; since September 2019)

*Email*: stefan.hammer@tugraz.at

*Mentors*: Peter Grabner, Birgit Vogtenhuber.

**Second phase of the doctoral program:****Christian Lindorfer**(Austria; since April 27, 2018)

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

*Mentors*: Robert Tichy, Florian Lehner.

*Thesis Title*: "Self-avoiding walks and formal languages".**Judith Kloas**(Germany; March 2014–March 2018)

*Email*: kloas@math.tugraz.at

*Mentors*: Mihyun Kang, Sebastian Müller.

*Thesis Title*: "Reflected and stopped random walks, and the distinguishing number of graphs".

*PhD Defense*: March 2, 2018.

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

*Examiners*: W. Imrich (Leoben), M. Peigné (Tours).

**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

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.

Showcases for possible PhD themes can be found here.