Schramm–Loewner Evolution

1. Front Matter

   Pages i-ix

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2. Introduction

   • Antti Kemppainen

   Pages 1-10

3. Introduction to Stochastic Calculus

   • Antti Kemppainen

   Pages 11-34

4. Introduction to Conformal Mappings

   • Antti Kemppainen

   Pages 35-47

5. Loewner Equation

   • Antti Kemppainen

   Pages 49-67

6. Schramm–Loewner Evolution

   • Antti Kemppainen

   Pages 69-100

7. Regularity and Convergence of Random Curves

   • Antti Kemppainen

   Pages 101-141

8. Back Matter

   Pages 143-145

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This book is a short, but complete, introduction to the Loewner equation and the SLEs, which are a family of random fractal curves, as well as the relevant background in probability and complex analysis. The connection to statistical physics is also developed in the text in an example case. The book is based on a course (with the same title) lectured by the author. First three chapters are devoted to the background material, but at the same time, give the reader a good understanding on the overview on the subject and on some aspects of conformal invariance. The chapter on the Loewner equation develops in detail the connection of growing hulls and the differential equation satisfied by families of conformal maps. The Schramm–Loewner evolutions are defined and their basic properties are studied in the following chapter, and the regularity properties of random curves as well as scaling limits of discrete random curves are investigated in the final chapter. The book is aimed at graduate students or researchers who want to learn the subject fairly quickly.

• Conformal Maps
• Loewner Equation
• Random Curves
• Scaling Limits
• SLE
• Convergence of Random Curves
• Area theorem
• Cardy-Smirnov formule
• Ising model
• Brownian motion
• Percolation
• Cardy formula
• Stochastic intergral
• Ito's formula
• Bessel process
• Poisson Kernel
• Schwarz-Christoffel mappings
• Loewner chains
• Schramm's principle

• Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

  Antti Kemppainen

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