Universität Duisburg-Essen
Fakultät Mathematik
Anita Winter

02nd of October 2019

Special Topic Course with integrated block seminar (2SWS+2SWS)
Winter Semester 2019/20
Tuesday, 14.15-15.45; WSC-S-U-3.02

Random walks on graphs and heat kernels estimates
(Vertiefungsbereich Stochastik bzw. Masterseminar)

The topic of random walks on graphs is vast one, and has close connections with many other areas of probability, as well as analysis, geometry and algebra. In the probabilistic direction, a random walk on a graph is just a reversible or symmetric Markov chain, and many results on random walks on graphs also hold for more general Markov chains. Iin this class the context will be restricted to random walks on graphs where each vertex has a finite number of neighbours. We are concerned with infinite graphs, and in particular those which have polynomial volume growth.

The first main topic is the relation between geometric properties of the graph and asymptotic properties of the random walk. A particular emphasis is on properties which are stable under minor perturbations of the graph, for example, the addition of a number of diagonal edges to the Euclidean lattice Z2. The precise definition of minor perturbation is given by the concept of a rough isometry, or quasi-isometry.

The second main theme of the class is deriving bounds on the transition density of the random walk, or the heat kernel, from geometric information on the graph. Once one has these bounds, many properties of the random walk can then be obtained in a straightforward fashion.

### Literature

• Martin Barlow "Random walks and heat kernels on graphs " (2017)

When does the class start?
The class starts on Tuesday, 15th of October.

What should I know if I want to participate?
This special topic course (Vertiefungsbereich Stochastik) is offered as part of the Master Program in Mathematics. Nevertheless, if you are interested in probability theory you are very welcome to participate. Preknowledge in probability theory (as covered by Wahrscheinlichkeitstheorie I) or Markov chains are required. Don't hesitate to send me an e-mail for more information:

How many credits do I get?
This class has a block seminar (Masterseminar) integrated. The topics will usually be papers which rely on the material presented in the class. It will take place in the end of the winter semester break. If you only take the class and pass an oral exam you get 3 CTS in Vertiefungsbereich Stochastik. Alternatively, you can present a paper in the block seminar. In that case you get 6 CTS either in Vertiefungsbereich Stochastik or as Master Seminar.

anita(dot)winter `at' due(dot)de