Discrete scale invariance and stochastic Loewner evolution

TL;DR

This paper introduces a new class of fractal curves with discrete scale invariance using the Loewner equation and WM functions, providing a method to classify such curves and analyzing their properties.

Contribution

It develops a framework to classify fractal curves with DSI via the Loewner equation and WM functions, linking fractal dimension to the diffusion coefficient.

Findings

Fractal dimension can be derived from the WM function's variance.

WM functions behave like Brownian motion in their trend.

Contour lines of 2D WM functions serve as physical examples of DSI curves.

Abstract

In complex systems with fractal properties the scale invariance has an important rule to classify different statistical properties. In two dimensions the Loewner equation can classify all the fractal curves. Using the Weierstrass-Mandelbrot (WM) function as the drift of the Loewner equation we introduce a large class of fractal curves with discrete scale invariance (DSI). We show that the fractal dimension of the curves can be extracted from the diffusion coefficient of the trend of the variance of the WM function. We argue that, up to the fractal dimension calculations, all the WM functions follow the behavior of the corresponding Brownian motion. Our study opens a way to classify all the fractal curves with DSI. In particular, we investigate the contour lines of 2D WM function as a physical candidate for our new stochastic curves.