Loewner Chains

TL;DR

This paper provides an introduction to stochastic Loewner evolutions and 2D growth processes, discussing their mathematical foundations, numerical simulations, and potential universality class distinctions.

Contribution

It offers a comprehensive overview of SLEs and 2D growth models, including new conjectures on their universality classes and detailed explanations of their mathematical descriptions.

Findings

Numerical simulations illustrate SLEs and 2D interfaces.

Schramm's argument links conformally invariant interfaces to SLEs.

A conjecture suggests Hele-Shaw and DLA may belong to different universality classes.

Abstract

These lecture notes on 2D growth processes are divided in two parts. The first part is a non-technical introduction to stochastic Loewner evolutions (SLEs). Their relationship with 2D critical interfaces is illustrated using numerical simulations. Schramm's argument mapping conformally invariant interfaces to SLEs is explained. The second part is a more detailed introduction to the mathematically challenging problems of 2D growth processes such as Laplacian growth, diffusion limited aggregation (DLA), etc. Their description in terms of dynamical conformal maps, with discrete or continuous time evolution, is recalled. We end with a conjecture based on possible dendritic anomalies which, if true, would imply that the Hele-Shaw problem and DLA are in different universality classes.