S4F1 Graduate Seminar in Probability:
Schramm-Loewner Evolution
Time: Wednesday 14st
Room: SR 0.007
Description:
Schramm-Loewner Evolution (or SLE) is the unique one-parameter family of random simple curves in the plane that are conformally invariant: If \(\Phi:D\to D'\) is a conformal mapping between two domains \(D,D'\subset\mathbb{R}^2\), then the law of the image of an SLE in \(D\) coincides with the law of the SLE in \(D'\).
This concept has received a lot of attention in the last decade and SLE is at the heart of some of the most fascinating recent developments in probability and mathematical physics.
On the one hand, the mathematics involved is a beautiful interplay of probability and complex analysis. On the other hand, these processes have opened to the door to a rigorous mathematical description of the scaling limits of many models from statistical mechanics at criticality in two dimensions. These include e.g. percolation, the Ising model and loop erased random walks.
In the seminar we will introduce the concept of Schramm-Loewner Evolution, study properties of the SLE processes and sketch the connection to scaling limits of lattice models.
Prerequisites:
solid background in stochastic analysis
General References:
- G. Lawler, Conformally Invariant Processes in the Plane, vol. 114 of Mathematical Surveys and Monographs, AMS
- W. Werner, Random planar curves and Schramm-Loewner evolutions, in Lectures on probability theory and statistics, vol. 1840 of Lecture Notes in Mathematics, Springer
- N. Berestycki and J. Norris, Lectures on Schramm-Loewner Evolution, Lecture notes [PDF]
- V. Beffara, Schramm-Loewner Evolution and other conformally invariant objects, in Probability and Statistical Physics in Two and More Dimensions, vol. 15 of Clay Mathematics Proceedings [PDF]
Schedule:
| Date | Time | Talk | Speaker |
|---|---|---|---|
| 11. Mai | 14:00 | Refresher on complex analysis | Lukas |
| 15:00 | Complex Brownian motion and potential theory | Alexander | |
| 01. June | 14:00 | Hulls and mapping out functions | Marius |
| 15:00 | Chordal Loewner chains | Claudio | |
| 08. June | 14:00 | Schramm–Loewner Evolution | Mathias |
| 15:00 | Phases of SLE | Manuel | |
| 15. June | 14:00 | Locality and restriction properties | Claudia |
| 22. June | 14:00 | Existence of the SLE curve | Nathalie |
| 15:00 | Fractal dimension of the SLE curve | Franca | |
| 29. June | 14:00 | Radial Schramm–Loewner Evolution | Georg |
| 06. July | 14:00 | SLE\(_4\) and the Gaussian free field | Benedikt |
| 13. July | 14:00 | Introduction to critical percolation | Katrin |
| 15:00 | Cardy's formula and Smirnov's theorem | Anastasiya | |
| 20. July | 14:00 | SLE\(_6\) and the percolation interface | Ewald |
| 20. July | 15:00 | One-arm exponent for critical percolation | Beni |