A short introduction to critical interfaces in 2D

Abstract

These notes are meant as a gentle introduction to the concepts of Loewner chains, local growth, Stochastic Loewner Evolutions (SLE) and their relationships to Conformal Field Theories (CFT). These concepts have played an important role in physics and mathematics during the recent years. The first section describes two discrete examples, the exploration process and loop-erased random walks. It can be read almost without any prerequisites. The aim is to show that even for curves defined purely in geometrical terms, it is useful to have a statistical mechanics viewpoint where the measure on curves is derived from a measure on local degrees of freedom of some model. A third model, diffusion-limited aggregation (DLA) is also introduced. The second section introduces Loewner chains and their relevance for the description of growth processes. A prerequisite is a minimal knowledge of complex analysis. The third section contains the derivation of the relevance of SLE in the description of interfaces when two properties, conformal invariance and the domain Markov property, are assumed/proved. The prerequisites are some knowledge of complex analysis and probability theory. The last section outlines the relation with Conformal Field Theory. This relation is via martingales and singular vectors. Again, the reader is assumed to have some basic knowledge of complex analysis and probability theory, and some familiarity with CFT. The discussion is informal. There is little or no claim at originality. We try to give some intuition based on explicit examples. Some physical applications are outlined. © 2012 Springer-Verlag Berlin Heidelberg.