2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety of 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 Schrammʹs 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 remarkable properties of SLE are explained, together with the tools for computing with it. This review has been written with the aim of filling the gap between the mathematical and the physical literature on the subject.