S4F2 - Graduate Seminar in Stochastic Analysis:

Gaussian Multiplicative Chaos and Liouville Quantum Gravity

SS20: Thursdays 10–12.   NB: The starting date is 28th May.

We will have two talks per week, each about 1 hour long. 

Due to the CoVid-19 pandemic, we must arrange the seminar via video (zoom). The link has been sent to the registered participans via email.


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 @



  • 28.05 (Nicolai Rohde): Intro, GFF and its properties [B: Chapters 1.2-1.6]
  • 28.05 (Min Liu): circle averages, thick points [B: Chapters 1.7-1.8] & [DS: Proposition 3.1] & [handout]
  • 04.06 (Marc Wedelstaedt): Liouville measure (LQG) in L^2 phase [B: Chapters 2.1-2.2] & [handout]
  • 04.06 (Ioannis Kavvadias): Liouville typical points, going beyond L^2 phase [B: Chapters 2.3-2.5] 
  • 18.06 (Marlene Rose): conformal covariance, relation to random surfaces [B: Chapters 2.6-2.7, parts of Chapter 5] and parts of [G]
  • 18.06 (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] 
  • 02.07 (Simon Schwarz): Conformal symmetry of extrema of the GFF [BL]
  • 09.07 (Marvin Bodenberger & Janis Papewalis): connections to random matrices [W]
  • 16.07 (Daria Frolova & Sid Maibach): Liouville QFT [RV: Chapter 3] & additional material


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

[B]: for which updated version appeared in March 2021:

and fill in details & discuss applications from additional material:







Further reading:

Basics on GFF & Liouville measure:

General GMC:

Applications - Random Matrices:

Applications - Random Surfaces and LQG:

Applications - Random Planar Maps:

Applications - Extrema of GFF:

Applications - Liouville Quantum Field Theory:

Applications - Finance: