Fractional Stochastic Loewner Evolution and Scaling Curves

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.

arXiv comment6 pages, 4 figures

References in corpus (15)

• Discrete scale invariance and complex dimensions
• Fractional differentiability of nowhere differentiable functions and dimensions
• The dimension of the SLE curves
• 2D growth processes: SLE and Loewner chains
• Conformal invariance in two-dimensional turbulence
• Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors
• Scale and conformal invariance in field theory: a physical counterexample
• Fracturing highly disordered materials
• Percolation with long-range correlated disorder
• Watersheds are Schramm-Loewner Evolution curves
• Fractional Brownian motion with a reflecting wall
• Conformal Curves on $W{O}_3$ Surface
• Localization in fractal and multifractal media
• Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation
• Self-organized criticality in cumulus clouds