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 @ hcm.uni-bonn.de

 

Schedule:

• 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]: http://www.math.stonybrook.edu/~bishop/classes/math638.F20/Berestycki_GFF_LQG.pdf for which updated version appeared in March 2021: http://homepage.univie.ac.at/nathanael.berestycki/Articles/master.pdf

and fill in details & discuss applications from additional material:

[BL]: http://arxiv.org/pdf/1410.4676.pdf

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

[RV]: http://arxiv.org/pdf/1602.07323.pdf

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

[G]: http://arxiv.org/pdf/1908.05573.pdf

 

Further reading:

Basics on GFF & Liouville measure:

• Sheffield: Gaussian free fields for mathematicians: http://arxiv.org/pdf/math/0312099.pdf
• Berestycki: An elementary approach to Gaussian multiplicative chaos: http://arxiv.org/pdf/1506.09113.pdf

General GMC:

• Rhodes & Vargas: Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity: http://arxiv.org/pdf/1602.07323.pdf
• Garban: Quantum gravity and the KPZ formula: http://arxiv.org/pdf/1206.0212.pdf
• Rhodes & Vargas: Gaussian multiplicative chaos and applications: http://arxiv.org/pdf/1305.6221.pdf
• Rhodes & Vargas: Lecture notes on Gaussian Multiplicative Chaos: http://www.newton.ac.uk/files/seminar/20150119100011001-297514.pdf
• Powell: Critical Gaussian Multiplicative Chaos: a rewiew: http://arxiv.org/abs/2006.13767

Applications - Random Matrices:

• Webb: The characteristic polynomial of a random unitary matrix and GMC, the L^2 phase: http://arxiv.org/pdf/1410.0939.pdf
• Nikula, Saksman & Webb: Multiplicative chaos and the characteristic polynomial of the CUE, the L^1 phase: http://arxiv.org/pdf/1806.01831.pdf
• Berestycki, Webb & Wong: Random Hermitian Matrices and Gaussian Multiplicative Chaos: http://arxiv.org/pdf/1701.03289.pdf
• Lambert, Ostrovsky & Simm: Subcritical multiplicative chaos for regularized counting statistics from random matrix theory: http://arxiv.org/pdf/1612.02367.pdf

Applications - Random Surfaces and LQG:

• Miller: Liouville quantum gravity as a metric space and a scaling limit: http://arxiv.org/pdf/1712.01571.pdf
• Gwynne & Miller: Existence and uniqueness of the Liouville quantum gravity metric for \gamma \in (0,2): http://arxiv.org/pdf/1905.00383.pdf
• Duplantier, Miller & Sheffield: Liouville quantum gravity as a mating of trees: http://arxiv.org/pdf/1409.7055.pdf

Applications - Random Planar Maps:

• Curien: A glimpse of the conformal structure of random planar maps: http://arxiv.org/pdf/1308.1807.pdf

Applications - Extrema of GFF:

• Biskup & Louidor: Conformal symmetries in the extremal process of two-dimensional discrete GFF: http://arxiv.org/pdf/1410.4676.pdf
• Biskup & Louidor: Extreme local extrema of two-dimensional discrete Gaussian free field: http://arxiv.org/pdf/1306.2602.pdf

Applications - Liouville Quantum Field Theory:

• David, Kupiainen, Rhodes & Vargas: Liouville Quantum Gravity on the Riemann sphere: http://arxiv.org/pdf/1410.7318.pdf
• Vargas: Lecture notes on Liouville theory and the DOZZ formula: http://arxiv.org/pdf/1712.00829.pdf
• Kupiainen, Rhodes & Vargas: Integrability of Liouville theory: proof of the DOZZ Formula: http://arxiv.org/pdf/1707.08785.pdf
• Guillarmou, Kupiainen, Rhodes & Vargas: Conformal bootstrap in Liouville Theory: http://arxiv.org/pdf/2005.11530.pdf

Applications - Finance:

• Bacry & Kozhemyak: Continuous cascade models for asset returns: http://www.cmap.polytechnique.fr/~bacry/ftp/papers/JEDC.pdf