Random Walk and Electrical Networks

In these lectures we will discuss “random walk on graphs”. These are general reversible Markov chains on at most countable state space in discrete and sometimes continuous time. One of our topics of interest will concern estimates on so-called heat kernels, that is, transition densities of such walks.

The course structure will be almost like a similar course done in ISI Bangalore by Siva Athreya. The course webpage can be found at http://www.isibang.ac.in/~athreya/Teaching/rgw/.

Note: Classes for 8th, 13th and 15th Feb will be taken by Ayan Bhattacharya.

References:

• Aldous and Fill- Reversible Markov chains and random walk on graphs. The on-line link is http://www.stat.berkeley.edu/~aldous/RWG/book.html.

• Martin Barlow. Random walk on graphs.

• Rick Durrett: Probability theory and examples. The on-line link is www.math.duke.edu/~rtd/PTE/PTE4_1.pdf.

• T. Kumagai: Random walk on disorder media and their scaling limits. The on-line link is www.kurims.kyoto-u.ac.jp/~kumagai/StFlour-TK.pdf

• Saloff-Coste: Lectures on finite Markov chains. On-line link is www. math.cornell.edu/~lsc/math778-only/stf.pdf.

• A.S. Sznitman: Lectures on Occupation times and Gaussian free field. On-line link is www.math.ethz.ch/u/sznitman/SpecialTopics.pdf

• Lyons and Peres: Probability on Trees and Networks. On-line link is  http://mypage.iu.edu/rdlyons/~prbtree/prbtree.html 

• Andras Telcs: Art of Random walk. http://www.springer.com/in/book/9783540330271

Examination: 25% midsem, 20% presentation, 55% end semester. Presentation topics will be given after mid semester exams.

Topics for presentation:

1. (Rohan)  Fenyman Kac formula and local time properties of RW: Theorem 1.13 (page 20) and Prop 1.14.

link: https://www.math.ethz.ch/content/dam/ethz/special-interest/math/department/Research/Research_Groups/Sznitman/Publications/2011_special_topics.pdf

2. (Sohom) Gaussian free field def and properties : Prop 2.1 and Prop 2.3

Same link above.

3. (Subhashis) Network reduction lemma - Paper by Ding, Lee and Peres, Lemma 2.9

Link : https://arxiv.org/abs/1004.4371

4. (Samriddha) Varopoulos Carne bound on heat kernel: Peres and Komajathy. Theorem 3.1

https://arxiv.org/abs/1506.04850

5.(Aniket) Isoperimetric inequality- Theorem 3.1 and Theorem 3.2 of Martin Barlow's notes. http://www.kurims.kyoto-u.ac.jp/~kumagai/rim10.pdf

6. (Soumya) Nash inequality and Theorem 3.6 of Martin Barlow's notes: same link above.

Assignments:

1. Assignment sheet 1.

2. Assignment sheet 2.

3. Assignment sheet 3.

4. Assignment sheet 4.