Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution.

Abstract

We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability p≥p(c). We numerically demonstrate that the scaling limit of planar LERW(p) curves, for all p>p(c), can be described by Schramm-Loewner evolution (SLE) with a single parameter κ 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 κ. Several geometrical tests are applied to ascertain this. All calculations are consistent with SLE(κ), where κ=1.732±0.016. This value of the diffusivity coefficient is outside the well-known duality range 2≤κ≤8. We also investigate how the winding angle of the LERW(p) crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p(c). 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.