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S4F2 - Graduate Seminar in Stochastic Analysis:

Gaussian Multiplicative Chaos and Liouville Quantum Gravity

Thursdays 10-12   in Endenicher Allee 60 - N 0.007 - Neubau 

 

The Gaussian Free Field (GFF) is a random "generalized" function, with mean zero and covariance given by the Green's function. It can be thought of as a universal object analogous to Brownian motion. In physics, it is also known as the "free bosonic field", which plays an important role in quantum field theory, quantum gravity, and statistical physics. The Liouville Quantum Gravity (LQG) is a random surface whose "Riemannian metric tensor'' can be expressed in terms of the exponential of the GFF, so that the "values" of the GFF determine volumes of domains on the surface. In physics, random surfaces are modelling gravity. Because the exponential of the GFF is not well-defined per se (as the GFF is only a "generalized" function), the LQG measure is defined via a limiting procedure. More generally, models for random surfaces can be obtained using the theory of Gaussian Multiplicative Chaos (GMC), which also has interesting connections to random matrix theory, turbulence, mathematical finance, etc.

 

Prerequisites: From Foundations in Stochastic Analysis: Brownian motion, martingales, uniform integrability. Some stochastic calculus is useful but not necessary. 

 

If you have any questions, please don't hesitate to contact me: 

firstname.lastname @ hcm.uni-bonn.de

 

Preliminary schedule (in progress):

Note that only the bold dates have been fixed. The other times might still be re-shuffled...

  • 09.04 (Nicolai Rohde): Intro, GFF and its properties [B: Chapters 1.2-1.6]
  • 16.04 (Min Liu): circle averages, thick points [B: Chapters 1.7-1.8] & [DS: Proposition 3.1] & [handout]
  • 23.04 (Marc Wedelstaedt): Liouville measure (LQG) in L^2 phase [B: Chapters 2.1-2.2] & [handout]
  • 30.04 (Ioauuis Kavvadias): Liouville typical points [B: Chapters 2.3-2.5] 
  • 07.05 (TBA): conformal covariance, relation to random surfaces [B: Chapters 2.6-2.7, parts of Chapter 5]
  • 14.05 (Simon Schwarz): TBA 
  • 28.05 (Aleksandra Korzhenkova): scaling relation for moments (multifractal spectrum, KPZ) [B: Chapters 3.1-3.3] & maybe [B: Chapters 3.4-3.5] & [RV: Chapter 2.3] 
  • 25.06: (Daria Frolova & Sid Maibach): Liouville QFT [RV: Chapter 3] & additional material (TBA)
  • 02.07: (Daria Frolova & Sid Maibach): Liouville QFT [RV: Chapter 3] & additional material (TBA)
  • 09.07: (Marvin Bodenberger & Janis Papewalis): connections to random matrices [W] & material from other papers (TBA)
  • 16.07: (Marvin Bodenberger & Janis Papewalis): connections to random matrices [W] & material from other papers (TBA)

Literature:

The plan is to start by following Berestycki's lecture notes:

[B]: http://www.statslab.cam.ac.uk/~beresty/Articles/oxford4.pdf

and fill in details & discuss applications from additional material:

[DS]: http://arxiv.org/pdf/0808.1560.pdf

[RV]: http://arxiv.org/abs/1602.07323

[W]: http://arxiv.org/pdf/1410.0939.pdf

[G]: http://arxiv.org/abs/1908.05573

 

Further reading: