Single random walker on disordered lattices (Q1072241)

Random walks on square lattice percolating clusters were followed for up to \(2\times 10^ 5\) steps. The mean number of distinct sites visited \(<S_ N>\) gives a spectral dimension of \(d_ s=1.30\pm 0.03\) consistent with superuniversality \((d_ s=4/3)\) but closer to the alternative \(d_ s=182/139\), based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a ''crossover'' from fractal-to-Euclidean behavior. Walks on two- and three- dimensional lattices are similar, except that those in three dimensions are more efficient.