A Guide to Stochastic Loewner Evolution and its Applications

TL;DR

This paper provides a comprehensive guide to Stochastic Loewner Evolution (SLE), explaining its mathematical foundation, properties, and applications in describing the scaling limits of critical models in statistical physics.

Contribution

It offers an accessible overview of SLE, connecting it to discrete models and including explicit proofs, thus serving as a valuable resource for researchers in the field.

Findings

SLE effectively describes the scaling limits of critical models.

Explicit proofs demonstrate key properties of SLE.

Connections between SLE and discrete models are elucidated.

Abstract

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 Loewner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Loewner'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.