Large deviations of Schramm-Loewner evolutions and Weil-Petersson Teichmuller space
Yilin Wang
The Loewner energy is a Mobius invariant deterministic quantity attached to a Jordan curve in the plane. It arises as the large deviation rate function for the Schramm-Loewner evolution (SLE) when the parameter goes to zero. A curve has finite Loewner energy if and only if it is a Weil-Petersson quasicircle, a class of curves that in the last decades has received significant attention by both physicists and mathematicians, but only very recently by probabilists. The Loewner-Kufarev energy is a dual quantity attached to certain continuous families of Jordan curves, which arises in the context of large deviations of SLEs when the parameter goes to infinity.
I will overview the basic knowledge of SLE, some results on the large deviation principles of SLEs, and background materials on Weil-Petersson quasicircles and Weil-Petersson Teichmuller space. I will show how ideas from random conformal geometry inspire results for Weil-Petersson quasicircles, in particular, through the example of SLE duality and the duality between Loewner and Loewner-Kufarev energy.
Prerequisite: Basic complex analysis, Brownian motion, basic notions of large deviation.
Main references:
- Yilin Wang: Large deviations of Schramm-Loewner evolutions: A survey. ArXiv:2102.07032
- Yilin Wang: Equivalent Descriptions of the Loewner Energy. Invent. Math., Vol. 218. 2, 573-621 (2019)
- Fredrik Viklund, Yilin Wang: The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles, arXiv:2012.05771 (2020)