Schramm–Loewner Evolution

• Main Author: Kemppainen, Antti (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2017.
• Series: SpringerBriefs in Mathematical Physics, 24
• Subjects: Physics.; Functions of complex variables.; System theory.; Probabilities.; Mathematical physics.; Mathematical Methods in Physics.; Mathematical Physics.; Probability Theory and Stochastic Processes.; Statistical Physics and Dynamical Systems.; Several Complex Variables and Analytic Spaces.; Complex Systems.

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...

Table of Contents:

• Introduction
• Iteration of conformal maps
• On stochastic models and connection to statistical physics
• An example: percolation model and Cardy’s formula
• On reading this book
• Introduction to stochastic calculus
• Brownian motion
• Stochastic integration
• Itô’s formula
• Further topics in stochastic calculus
• Conformal invariance of two-dimensional Brownian motion
• Weak convergence of probability measures
• Introduction to conformal mappings
• Harmonic functions
• Conformal maps
• From Area theorem to distortion
• Conformally invariant tools
• Loewner equation
• Conformal maps of the upper half-plane
• Loewner chains
• Loewner equations in D and Sp
• Schramm–Loewner evolution.-Schramm–Loewner evolution and its elementary properties
• Advanced properties of SLE
• Proofs for some of the advanced properties
• Variants of SLE
• Moments of the derivative of the Loewner map of SLE(k)
• Regularity and convergence of random curves
• Continuity properties of the Loewner chains
• Continuity of SLE(k)
• Convergence of interfaces in the site percolation model
• Index.