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

Abstract

We study the scaling limit of planar loop erased random walk (LERW) on the percolation cluster, with occupation probability $p\geq p_c$. 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 $κ$ which is close to normal LERW in 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 $\mathrm{SLE}_κ$, where $κ=1.732\pm0.016$. This value of the diffusivity coefficient is outside of the well-known duality range $2\leq κ\leq 8$. We also investigate how the winding angle of the LERW$_p$ crosses over from {\it Euclidean} to {\it 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. We believe that this finding should, to some degree, help us to understand and predict the existence of conformal invariance in disordered and fractal landscapes.