  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.

Statistical Mechanics