# Graduate Seminar Probability Theory S4F1, SS 2013

Self-avoiding walks and non-Markovian random structures

**Time and Place: Thursdays, 14-16h, 0.006, Mathematics Center **

**Preparatory meeting: Wednesday, February 6, 16h, in room 3.037 **

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**Those interested in participating (in particularly those who cannot come on the 6th), please send me an email!!!**

Topic:

The classical non-Markovian random geometric structure is the self-avoiding random walk. As simple as it is to describe, analysing its large-scale properties is a highly non-trivial problem. Methods (lace expansion) have been devised to deal with it in high enough dimensions, where the constraint of self-avoidance becomes irrelevant, and recently methods from conformal analysis have been used to tackle the two-dimensional case. In this seminar we pick up some of these developments. The basic text will be the Clay lectures notes by Bauerschmidt et al. There are a number of further notes and papers by Gordon Slade (lace expansions) and Duminil-Copin and Smirnov. They use Smirnov's approach of discrete complex analysis, which is quite a fascinating topic in itself. Depending on the participants interest, we could also go into this topic more deeply.

### Further reading:

- Duminil-Copin, Smirnov, "The connection constant on the noneycomb lattice"
- Smirnov's ICM Proceedings on Discrete COmplex Analysis
- Duminil-Copin and Smirnov, Conformal invariance of lattice models
- Brydges, Imbrie, Slade, Functional Integral Representation of SAW