# Schramm–Loewner Evolution

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 24)

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Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 24)

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

- DOI https://doi.org/10.1007/978-3-319-65329-7
- Copyright Information The Author(s) 2017
- Publisher Name Springer, Cham
- eBook Packages Physics and Astronomy
- Print ISBN 978-3-319-65327-3
- Online ISBN 978-3-319-65329-7
- Series Print ISSN 2197-1757
- Series Online ISSN 2197-1765
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