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

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.