Gaussian free field and Liouville quantum gravity

SS 2021

General Information


Online via Zoom :

Password: GFF-LQG

Monday 14:00-16:00
Wednesday 14:00-16:00

First lecture: April 12, 2021

Lecturer:  Prof. Dr. Chiranjib Mukherjee
Assistance:  Konstantin Recke

This lecture in the course overview

These tutorials in the course overview

Course syllabus:

In the first part of the course, we will start with an introduction to the Gaussian free field (GFF), which is an object which has been at the heart of some recent groundbreaking developments in probability theory and its connections to many other branches of mathematics (e.g. complex analysis, geometry, partial differential equations) and statistical physics. Loosely speaking, GFF is a collection of Gaussian random “variables“ indexed by a two-dimensional domain with a prescribed covariance structure.

We will then discuss a few applications to the emerging theory of Liouville quantum gravity (LQG), which provides the most natural way of choosing a two-dimensional random surface (just like, if we are given a finite set X, the easiest way to choose a random element of X is uniformly, i.e. by assigning equal probability to each element of X). Mathematically, LQG is related to the problem of giving a meaning to the "exponential of the Gaussian free field". We will then see how LQG appears as the scaling limit of random planar maps (just like Brownian motion appears as the scaling limit of simple random walks, and many other discrete models).

The course will be kept at a very basic level and would require only some background from measure-theoretic probability. Further necessary material (e.g. tools from complex analysis and advanced probability) will be discussed in the lecture.

Literature: See Learnweb.

Please enroll in the Learnweb course for this lecture.

Course assessment: Successful completion of 40% of the exercise sheets as well as an oral/written exam at the end of the course (format, date and time tba.).

Online via Zoom:

Password: GFF-LQG_2

Monday 12:00-14:00

The exercise sheets can be found in the Learnweb course and below:

Exercise Sheet 1Exercise Sheet 2Exercise Sheet 3
Exercise Sheet 4Exercise Sheet 5Exercise Sheet 6
Exercise Sheet 7Exercise Sheet 8Exercise Sheet 9
Exercise Sheet 10