GFF and LQG (2018)

Advanced topic in Stochastic Analysis: Gaussian Free Field and Liouville Quantum Gravity.

Location: TU Berlin, Mathematics building, room: MA 645

Time: Winter semester 2018. Mondays 12-14, room MA 645

Sprechzeiten: 10-12 Dienstage oder Verabredung.

Class webpage: https://sites.google.com/site/tranvohuy/reading-seminars/dgff-gff-and-lqg-2018

Language: English.

Background: Knowledge of Brownian motion, linear algebra, linear analysis, complex analysis is very useful. Most of the statements in the course have analogies to Brownian motion.

Short description: Our main goal is to study the Imaginary Geometry theory, developed by Jason Miller, Scott Sheffield, and others. This is a fast evolving theory that describes random surfaces with random measures and metrics which are conjectured and proven to be the scaling limits of many discrete models in statistical physic. This theory is studied extensively but still needs to understand from many angles.

Long description: See here https://goo.gl/1JLfp7

Summary of content:

• • We want to understand the following inspiring pictures: GFF, LQG, flow lines.

  • We mainly follow Berestycki's lecture note.

  • First part is about Discrete Gaussian Free Field. DGFF is a discrete version of GFF, easier to understand. So we will learn it carefully with the analogies of GFF in mind. Easy for experts, great for students.

  • The second part is about Gaussian Free Field/ Gaussian Field. Easy for experts, good for students.

  • The third part is about Liouville quantum gravity (or Gaussian chaos). We will study the random measures (chaos) and its relation to the underlying Gaussian field. Interesting for experts.

  • The last part is about the relation between GFF, LQG, and Schramm Loewner equation (SLE). Interesting and confusing for the audience and the lecturer.

Tentative and naive material

DGFF:

• • Multivariate Gaussian random variables

  • Discrete Laplacian, discrete Green functions

  • Definition and equivalent definitions of DGFF

  • Different versions

  • Properties

  • Simulations

Space of generalized functions, Sobolev spaces:

• • Schwarz space

  • Laplace eq., Green functions

  • Sobolev spaces, Hilbert spaces, ONB

GFF:

• • General theory of Gaussian fields

  • Defs, equi. defs of GFF

  • Properties: circle average, strong/weak Markov properties, ...

  • Diff. versions: GFF free boundary, mixed boundary, etc.

LQG, or Gaussian chaos:

• • Existence theorem.

  • Uniqueness of the random measure

  • Properties: KPZ formula, continuity in gamma, support of LQG measures

  • Def of quantum surfaces: quantum cone, quantum wedge.

Quantum surfaces and SLE:

• • SLE calculus.

  • Coupling theorem between SLE and GFF.

  • Flow lines.

  • Understand the statements in the paper "Imaginary geometry I"

  • Local sets.

  • Counter flowlines

Presentations for students:

• • Berestycki, Powell, Ray, a characterization for GFF, slide, video.

  • Berestycki, Jackson, Rohde-Schramm theorem via GFF

  • Junnila, Saksman, The uniqueness of the Gaussian chaos revisited.

  • Exercises.

  • Guillaume Remy, Fyodorov-Bouchaud formula, slide

  • Stéphane Benoist: Natural parametrization of SLE: GFF point of view

References:

• • Nathanael Berestycki's lecture note, papers

  • Wendelin Werner's lecture note

  • Fractional Gaussian Fields, a survey. This is a good source to define GFF rigorously. I should have started with this note.

  • Eero Saksman's papers

  • Jason Miller and Scott Sheffields and Betrand Duplantier

  • Stéphane Benoist's note.

  • Jason Miller's lecture note.

Announcement:

-12.10.2018: no class on 29.10 since I will be in Oxford, UK.

First set of lecture notes: DGFF, or Overleaf for the latest (choose the tex file, then hit "Recompile").

-26.11: second set of lecture notes: Continuous GFF (not finished), or click overleaf (on left-side panel, click on "Continuous_GFF.tex", then on the right-side panel, click "Recompile")

-5.1.2019: Unfortunately, there is NO lecture on 7.1 and 14.1.