Random planar curves and Schramm-Loewner evolutions

TL;DR

This paper reviews recent advances in understanding two-dimensional random curves, focusing on conformal invariance, scaling limits, and the properties of Schramm-Loewner evolutions (SLE), including their applications to various models.

Contribution

It provides an overview of the development, properties, and applications of SLE, highlighting its role in analyzing critical phenomena in two-dimensional systems.

Findings

Definition and properties of SLE are established.

SLE is used to compute probabilities and critical exponents.

Connections between SLE and models like Brownian motion, percolation, and spanning trees are demonstrated.

Abstract

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.