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We study loop erased random walk (LERW) on the percolation cluster, withoccupation probability $p\geq p_c$, in two and three dimensions. We find thatthe fractal dimensions of LERW$_p$ is close to normal LERW in Euclideanlattice, for all $p>p_c$. However our results reveal that LERW on criticalincipient percolation clusters is fractal with $d_{f}=1.217\pm0.0015$ for d = 2and $1.44\pm0.03$ for d = 3, independent of the coordination number of thelattice. These values are consistent with the known values for optimal pathexponents in strongly disordered media. We investigate how the behavior of theLERW$_p$ crosses over from Euclidean to fractal geometry by graduallydecreasing 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 opensup a new theoretical window regarding diffusion process on fractal and randomlandscapes.

Loop erased random walk on percolation cluster: Crossover from Euclidean to fractal geometry