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

TL;DR

This paper investigates the scaling limit of loop erased random walks on percolation clusters, revealing a unique SLE parameter at criticality and exploring the transition from Euclidean to fractal geometry.

Contribution

It demonstrates that LERW on percolation clusters at criticality is compatible with SLE but with a different diffusivity coefficient, expanding understanding of conformal invariance in disordered systems.

Findings

LERW on percolation clusters for p > p_c is described by SLE with a specific κ.

At criticality, LERW is compatible with SLE with κ=1.732±0.016, outside the usual duality range.

The study identifies crossover exponents and scaling relations for the transition from Euclidean to fractal geometry.

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 $\kappa$ 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 $\kappa$. Several geometrical tests are applied to ascertain this. All calculations are consistent with $\mathrm{SLE}_{\kappa}$, where $\kappa=1.732\pm0.016$. This value of the diffusivity coefficient is outside of the well-known duality range $2\leq \kappa\leq 8$. 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.