Abstract
We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability . We numerically demonstrate that the scaling limit of planar curves, for all , can be described by Schramm-Loewner evolution (SLE) with a single parameter that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient . Several geometrical tests are applied to ascertain this. All calculations are consistent with , where . This value of the diffusivity coefficient is outside the well-known duality range . We also investigate how the winding angle of the crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter from 1 to . For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.
- Received 2 June 2014
DOI:https://doi.org/10.1103/PhysRevE.90.022129
©2014 American Physical Society