The Loewner equation: maps and shapes

TL;DR

This paper discusses recent advances in understanding fractal shapes in two dimensions through Schramm-Loewner evolution (SLE), linking complex analysis and probability to critical phenomena and percolation.

Contribution

It introduces the SLE approach as a novel method for analyzing fractal shapes in critical phenomena, bridging analytic function theory and probability.

Findings

SLE provides a new framework for calculating fractal shapes.

The approach connects complex analysis with statistical physics.

It offers insights into critical phenomena and percolation models.

Abstract

In the last few years, new insights have permitted unexpected progress in the study of fractal shapes in two dimensions. A new approach, called Schramm-Loewner evolution, or SLE, has arisen through analytic function theory and probability theory, and given a new way of calculating fractal shapes in critical phenomena, the theory of random walks, and of percolation. We present a non-technical discussion of this development aimed to attract the attention of condensed matter community to this fascinating subject.