Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation

TL;DR

This paper links anisotropic percolation boundaries to Stochastic Loewner Evolution driven by anomalous Brownian motions, revealing how different diffusion types influence fractal anisotropy at criticality.

Contribution

It demonstrates that the perimeters of certain percolation clusters are scaling limits of SLE driven by anomalous diffusion, providing a new perspective on non-Markovian processes.

Findings

Perimeters of multi-layered and directed percolation are linked to anomalous Brownian motion in SLE.

Long-range correlated time series produce anisotropic traces, supporting the connection.

The study offers new insights into the mathematical and physical interpretation of non-Markovian processes.

Abstract

We disclose the origin of anisotropic percolation perimeters in terms of the Stochastic Loewner Evolution (SLE) process. Precisely, our results from extensive numerical simulations indicate that the perimeters of multi-layered and directed percolation clusters at criticality are the scaling limits of the Loewner evolution of an anomalous Brownian motion, being subdiffusive and superdiffusive, respectively. The connection between anomalous diffusion and fractal anisotropy is further tested by using long-range power-law correlated time series (fractional Brownian motion) as driving functions in the evolution process. The fact that the resulting traces are distinctively anisotropic corroborates our hypothesis. Under the conceptual framework of SLE, our study therefore reveals new perspectives for mathematical and physical interpretations of non-Markovian processes in terms of anisotropic paths at criticality and vice-versa.