Uni Bonn SS20:
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 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.
- 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
Literature:
The plan is to start by following Berestycki's lecture notes:
[B]: here and updated version (appeared in March 2021) ...
... and fill in details & discuss applications from additional material:
[BL]: Marek Biskup & Oren Louidor, Conformal symmetries in the extremal process of two-dimensional discrete Gaussian free field.
[DS]: Bertrand Duplantier and Scott Sheffield, Liouville Quantum Gravity and KPZ.
[RV]: Remi Rhodes and Vincent Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity.
[W]: Christian Webb, The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos.
[G]: Ewain Gwynne, Random Surfaces and Liouville Quantum Gravity.
Further reading:
Basics on GFF & Liouville measure:
General GMC:
Rhodes & Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
Rhodes & Vargas, Gaussian multiplicative chaos and applications
Rhodes & Vargas, Lecture notes on Gaussian Multiplicative Chaos
Applications - Random Matrices:
Webb, The characteristic polynomial of a random unitary matrix and GMC, the L^2 phase
Nikula, Saksman & Webb, Multiplicative chaos and the characteristic polynomial of the CUE, the L^1 phase
Berestycki, Webb & Wong, Random Hermitian Matrices and Gaussian Multiplicative Chaos
Lambert, Ostrovsky & Simm, Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
Applications - Random Surfaces and LQG:
Miller, Liouville quantum gravity as a metric space and a scaling limit
Gwynne & Miller, Existence and uniqueness of the Liouville quantum gravity metric for \gamma \in (0,2)
Duplantier, Miller & Sheffield, Liouville quantum gravity as a mating of trees
Applications - Random Planar Maps:
Applications - Extrema of GFF:
Biskup & Louidor, Conformal symmetries in the extremal process of two-dimensional discrete GFF
Biskup & Louidor, Extreme local extrema of two-dimensional discrete Gaussian free field
Applications - Liouville Quantum Field Theory:
David, Kupiainen, Rhodes & Vargas, Liouville Quantum Gravity on the Riemann sphere
Vargas, Lecture notes on Liouville theory and the DOZZ formula
Kupiainen, Rhodes & Vargas, Integrability of Liouville theory: proof of the DOZZ Formula
Guillarmou, Kupiainen, Rhodes & Vargas, Conformal bootstrap in Liouville Theory
Applications - Finance:
- Bacry & Kozhemyak, Continuous cascade models for asset returns