## V5F3 - Advanced Topics in Stochastic Analysis:

## Introduction to Schramm-Loewner Evolution

**WS19/20: Mondays 12–14 and Thursdays 8–10. ** Endenicher Allee 60 - SemR 1.008

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Schramm-Loewner Evolution (SLE) is a key concept when studying the geometry of random structures. It lies at the interface between probability, geometry, and analysis, combining beautiful theory of conformal mappings to stochastic analysis and properties of Brownian motion. The SLE is a random fractal curve in the plane, whose most famous applications include the understanding of geometric properties of statistical models in two dimensions (random walks, percolation, Ising model, Gaussian free field, ...).

The goal of the course is to provide an introduction to the definition, properties, and applications of the SLE.

We will cover some background material on complex analysis and stochastic analysis, when needed (the precise plan will depend on the participants’ background knowledge and wishes). Along the way, we develop Loewner’s theory for growth processes encoded in conformal maps. Then we define the SLE processes and prove their basic properties. Time permitting, we discuss further properties and applications of the SLE.

## Course Log:

Here appear rough descriptions of the past (and following) lectures.

- 07.10: Overview presentation, practicalities (for the slides, please contact me by email)
- 10.10: Refreshing some basics in complex analysis: holomorphic/analytic maps, their properties, maximum modulus principle, Schwarz lemma
- 14.10: Complex analysis continues: conformal maps, Möbius transformations, Riemann mapping thm, SLE(0)
- 17.10: Area theorem, Koebe 1/4 theorem, Koebe distortion theorem, ...
- 21.10: Boundary properties of conformal maps, half-plane capacity, ...
- 24.10: Recap of some notions from stochastic analysis. Conformal invariance of 2D Brownian motion, Kakutani's formula and harmonic functions, Beurling estimate
- 28.10: Continuity and positivity of half-plane capacity, first steps towards Loewner theory of growth processes
- 31.10: Discussion on Exercise Sets 1,2, and 3 (about complex analysis)
- 04.11: Loewner's theorem, idea of the proof, examples, and consequences
- 07.11: proof of Loewner's theorem
- 11.11: finishing the proof of Loewner's theorem, definition of SLE and first properties
- 14.11: Schramm's theorem, ideas from discrete models
- 18.11: Bessel flow associated to SLE, SLE hitting the real line, phases of SLE
- 21.11: Discussion on Exercise Sets 4,5, and 6
- 25.11: transience of the SLE curve when 0 ≤ kappa < 8
- 28.11: SLE swallowing points in the interior, transience of the space-filling SLE curve with kappa ≥ 8
- 02.12: Schramm's formula, SLE Green's function (one-point function), sketch why SLE is a curve
- 05.12: Brownian excursions, comparing Loewner evolution in different domains
- 09.12: SLE in other domains, locality property (kappa = 6), Cardy's formula
- 12.12: change of target point for SLE, SLE(kappa, rho), tilting SLE by (local) martingales involving Brownian loop measure
- 16.12: probability of SLE avoiding a hull, conformal restriction property (kappa = 8/3)
- 17.12: Discussion on Exercise Sets 7,8, and 9
- 19.12: conformal restriction measures and related topics
- 13.01: Definition and main properties of the Gaussian free field (GFF): discrete case
- 16.01: GFF on planar domains as a random distribution, level lines of the GFF as SLE(4) curves
- 20.01: proof of the GFF - SLE(4) coupling, Step 1: the coupling
- 23.01: proof of the GFF - SLE(4) coupling, Step 2: SLE(4) is determined by the GFF
- 27.01: further relations of the GFF and SLE
- 30:01: Discussion on Exercise Sets 9,10, and 11

## Exercises:

- Exercise Set 1
- Exercise Set 2
- Exercise Set 3
- Exercise Set 4
- Exercise Set 5
- Exercise Set 6
- Exercise Set 7
- Exercise Set 8
- Exercise Set 9
- Exercise Set 10
- Exercise Set 11

If you have any questions, please don't hesitate to contact me:

*firstname.lastname @ hcm.uni-bonn.de*

## Literary Remarks:

We are not following any source directly. Below appear some pointers to find most of the material covered so far. (In the lectures, some additional results might have been discussed and references were given there.)

background:**Stochastic analysis**- We won't go through basic Stochastic analysis in these lectures but we'll state all the results that we need.
*Chapter 2 in [Kem]*contains the necessary background for us.- Further reading:
*Chapter 1 in [Law]*is quite comprehensive.

background:*Complex analysis**Chapter 3 in [Kem]:*has not many details but is rather quick to look at.- Further reading: the textbooks
*[Con]*are quite easy to read. See also*[Ahl].*

**Complex Brownian motion:***Chapter 2 and Chapter 3.1 in [BN]*- Further reading:
*Chapters 2.1-2.3 in [Law]*

**Koebe distortion theory:***Chapters 3.1-3.2 in [Law]*- Further reading:
*Chapter 3.3 in [Kem]*

*Half-plane capacity:**Chapter 4.1 in [Kem]*- Further reading:
*Chapter 3.4 in [Law]*

**Loewner differential equation:***Chapter 4.2 in [Kem]*- Further reading:
*Chapter 4.1 and Chapter 4.4 in [Law]*

**SLE -- definition and first properties:***Chapter 5.1 in [Kem]*- Further reading:
*Chapter 6.1 in [Law]*

**SLE -- phases and hitting probabilities of the real line R:***Chapter 10 in [BN]*has the Bessel flow and hitting probabilities of R for all kappa.*Chapter 11 in [BN]*has the phases of SLE(kappa) for 0 ≤ kappa < 8, and transience of the curve for 0 ≤ kappa < 8 excluding kappa = 4. (Proposition 6.12 in [Law] has transience of the curve for kappa = 4.)- Further reading:
*Chapter 6.2 in [Law]*and the original paper of Rohde and Schramm <http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf>

**SLE -- hitting and swallowing probabilities inside the upper half-plane:**- Lawler's new book:
*Chapter 2 in *<http://www.math.uchicago.edu/~lawler/slepaper.pdf>**** - Further reading:
*Chapter 7 in [Law]*and*Chapters 5.3.4 - 5.3.6 in [Kem]* - See also
*Sections 6-7*in the original paper of Rohde and Schramm <http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf>

- Lawler's new book:
**comparing SLE in different domains, locality:**- Lawler's new book:
*Chapter 5 in *<http://www.math.uchicago.edu/~lawler/slepaper.pdf>**** - Further reading:
*Chapter 4.5, Chapter 6.3, and Chapter 6.7-6.8 in*[Law]

- Lawler's new book:
*SLE(6) and Cardy's formula:**Chapters 6.3 and 6.8 in [Law]*- Further reading:
*Chapter 13 in [BN]*

*SLE(8/3) and conformal restriction:**Chapter 6.4 in [Law]*- Further reading:
*Chapter 14 in [BN]*

*Conformal restriction measures:**Chapter 9 in [Law] and Chapter 4 in <**http://projecteuclid.org/download/pdfview_1/euclid.ps/1444653628*>- Further reading:
- the original paper of Lawler, Schramm, and Werner: <http://www.ams.org/journals/jams/2003-16-04/S0894-0347-03-00430-2/S0894-0347-03-00430-2.pdf>
- the lecture notes of Wu: <http://projecteuclid.org/download/pdfview_1/euclid.ps/1444653628>

*Gaussian free field (GFF):**Chapters 1.1 and 2.1*in*<**http://pdfs.semanticscholar.org/2607/b47d11a2b1758063795bb33348d9f963011d.pdf*>- There is a updated version of the lecture notes here: <
>*http://arxiv.org/abs/2004.04720* - Further reading:
*Chapter 15 and Chapter 3.3 in [BN]*

*GFF - SLE(4) "level line" coupling:**Chapter 3.1*in*<**http://pdfs.semanticscholar.org/2607/b47d11a2b1758063795bb33348d9f963011d.pdf*>- There is a updated version of the lecture notes here: <
>*http://arxiv.org/abs/2004.04720* - Further reading:
*Chapter 15.4 in [BN]*

*GFF - SLE(\kappa) "flow line" coupling:*- Miller & Sheffield, Imaginary geometry I <http://arxiv.org/pdf/1201.1496.pdf>

*GFF - SLE(\kappa) "zipper" coupling:**Chapter 6*in N. Berestycki's lecture notes*<**http://www.statslab.cam.ac.uk/~beresty/Articles/oxford4.pdf*>- Further reading: Sheffield's original paper <http://arxiv.org/abs/1012.4797>

[Ahl]: Lars Ahlfors. Complex Analysis.

[Con]: John Conway. Functions of One Complex Variable, Volumes 1-2

[BN]: Nathanael Berestycki and James Norris. *Lecture notes on SLE*

[Kem]: Antti Kemppainen. *Schramm-Loewner evolution*

[Law]: Gregory Lawler. *Conformally Invariant Processes in the Plane*

[Wer]: Wendelin Werner. Random planar curves and Schramm-Loewner evolutions.