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A guide to stochastic Löwner evolution and its applications. (English) Zbl 1157.82327
Summary: This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Löwner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Löwner’s method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.

MSC:
82B27 Critical phenomena in equilibrium statistical mechanics
30C55 General theory of univalent and multivalent functions of one complex variable
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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