Fractional Stochastic Loewner Evolution and Scaling Curves

TL;DR

This paper introduces fractional stochastic Loewner evolution (FSLE), a novel extension of Schramm's stochastic Loewner equation, enabling classification of fractal curves based on the Hurst index, with applications in statistical mechanics.

Contribution

The paper proposes FSLE, incorporating fractional time series and local fractional operators, to generate and classify a broader class of fractal curves.

Findings

FSLE generates diverse fractal curves with varying Hurst indices.

FSLE curves exhibit differences in fractal properties beyond fractal dimension.

The formalism is applicable to various two-dimensional phenomena in physics and nature.

Abstract

The Stochastic Loewner equation, introduced by Schramm, gives us a powerful way to study and classify critical random curves and interfaces in two-dimensional statistical mechanics. New kind of stochastic Loewner equation, called fractional stochastic Loewner evolution (FSLE), has been proposed for the first time. Using the fractional time series as the driving function of the Loewner equation and local fractional integrodifferential operators, we introduce a large class of fractal curves. We argue that the FSLE curves, besides the fractal dimension calculations, have crucial differences which caused by the Hurst index of the driving function. This extension opens a new way to classify different types of scaling curves based on the Hurst index of the corresponding driving function. Such formalism appear to be suitable to deal with the study of a wide range of two-dimensional curves appearing in statistical mechanics or natural phenomena.