  We study loop erased random walk (LERW) on the percolation cluster, with
occupation probability $p\geq p_c$, in two and three dimensions. We find that
the fractal dimensions of LERW$_p$ is close to normal LERW in Euclidean
lattice, for all $p>p_c$. However our results reveal that LERW on critical
incipient percolation clusters is fractal with $d_{f}=1.217\pm0.0015$ for d = 2
and $1.44\pm0.03$ 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 new theoretical window regarding diffusion process on fractal and random
landscapes.
