Probability and Stochastic Analysis - University of Bonn

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:

DateTimeTalkSpeaker
11. Mai14:00Refresher on complex analysisLukas
15:00Complex Brownian motion and potential theoryAlexander
01. June14:00Hulls and mapping out functionsMarius
15:00Chordal Loewner chainsClaudio
08. June14:00Schramm–Loewner EvolutionMathias
15:00Phases of SLEManuel
15. June14:00Locality and restriction propertiesClaudia
22. June14:00Existence of the SLE curveNathalie
15:00Fractal dimension of the SLE curveFranca
06. July14:00SLE$_4$ and the Gaussian free fieldBenedikt
20. July14:00SLE$_6$ and the percolation interfaceEwald