Stochastic Loewner Evolution

TL;DR

Stochastic Loewner evolution (SLE) is a mathematical tool that generates and analyzes scale-invariant fractal curves in two dimensions, linking conformal invariance with stochastic processes.

Contribution

This paper provides a simple, heuristic overview of SLE, highlighting its role in understanding critical phenomena and fractal curves in statistical physics.

Findings

SLE generates scale-invariant fractal curves in the complex plane.

The fractal dimension of SLE curves is determined by Brownian motion strength.

SLE bridges lattice models and conformal field theory in critical phenomena.

Abstract

Stochastic Loewner evolution also called Schramm Loewner evolution (abbreviated, SLE) is a rigorous tool in mathematics and statistical physics for generating and studying scale invariant or fractal random curves in two dimensions. The method is based on the older deterministic Loewner evolution introduced by Karl Loewner, who demonstrated that an arbitrary curve not crossing itself can be generated by a real function by means of a conformal transformation. In 2000 Oded Schramm extended this method and demonstrated that driving the Loewner evolution by a one-dimensional Brownian motion, the curves in the complex plane become scale invariant; the fractal dimension turns out to be determined by the strength of the Brownian motion. SLE fills a gap in our understanding of the critical properties of a variety of lattice models in their scaling limits and supplements the result obtained by means of conformal field theory. In this paper we attempt to provide a simple and heuristic discussion of some of the important aspects of SLE.