2D growth processes: SLE and Loewner chains

TL;DR

This review introduces 2D growth processes, focusing on stochastic Schramm-Loewner evolutions (SLE), explaining their connection to critical systems in statistical mechanics and detailing their properties and computational tools.

Contribution

It provides a comprehensive overview linking 2D growth processes, especially SLE, with statistical mechanics, bridging mathematical and physical perspectives.

Findings

SLE describes interfaces in 2D critical systems.

Connections between statistical mechanics and SLE are elucidated.

Properties and tools for SLE computation are explained.

Abstract

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.