Loop-erased random walk on a percolation cluster: crossover from Euclidean to fractal geometry.

Abstract

We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ≥ p_{c}, in two and three dimensions. We find that the fractal dimensions of LERW_{p} are close to normal LERW in a Euclidean lattice, for all p>p_{c}. However, our results reveal that LERW on critical incipient percolation clusters is fractal with d_{f}=1.217 ± 0.002 for d=2 and 1.43 ± 0.02 for d=3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior 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 work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.