C8.5 Introduction to Schramm-Loewner Evolution - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Dmitry Belyaev
General Prerequisites: 

Analysis: Basic knowledge of differential equations, measure and integration, basic complex analysis, conformal map theory (you might consider taking the C4.8 course in the first term or read the lecture notes).
Probability: Martingales, Itô formula.
Some knowledge about lattice models such as percolation, loop-erased random walk, Ising model etc. will be beneficial but not required. All the necessary parts will be covered in the lectures.

Course Term: 
Hilary
Course Lecture Information: 

16 lectures.

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

The Schramm-Loewner Evolution (SLE) was introduced in 1998 in order to describe all possible conformally invariant scaling limits that appear in many lattice models of statistical physics. Since then the subject has received a lot of attention and developed into a thriving area of research in its own right which has a lot of interesting connections with other areas of mathematics and physics. Beyond the aforementioned lattice models it is now related to many other areas including the theory of `loop soups', the Gaussian Free Field, and Liouville Quantum Gravity. The emphasis of the course will be on the basic properties of SLE and how SLE can be used to prove the existence of a conformally invariant scaling limit for lattice models.

Learning Outcomes: 

The students will learn basic properties of SLE and understand the main techniques used to do SLE computations. They will learn what it means for a random curve to converge to SLE and about a possible way to prove this convergence.

Course Synopsis: 

1) (2 lectures) A quick recap of the necessary background from complex analysis. We will go through the Riemann mapping theorem and basic properties of univalent functions both in the unit disc and in the upper half plane. In these lectures I will give the main results and connections between them but not the proofs.

2) (4 lectures) Half-plane capacity, Beurling estimates and (deterministic) Loewner Evolution. We will show that any 'nice' curve can be described by a Loewner Evolution and will study the main properties of a Loewner Evolution driven by a measure.

3) (6 hours) Defi nition of Schramm-Loewner Evolution and Schramm's principle stating that SLEs are the only conformally invariant random curves satisfying the so called 'domain Markov property'. We will study the main properties of SLE. In particular, we will study its phase transitions and two special cases when SLE has the 'locality' property and the 'restriction' property.

4) (4 lectures) Show the crossing probability for the critical percolation on the triangular lattice has a conformally invariant scaling limit (Cardy's formula). Prove that this implies that the percolation interfaces converge to SLE curves.

Reading List: 

1. G.F. Lawler, Conformally invariant processes in the plane (No. 114). American Mathematical Soc. 2008.

2. Antti Kemppainen, Schramm-Loewner Evolution, Schramm-Loewner Evolution. Springer, Cham, 2017. 69-100.