An efficient implementation of the exact enumeration method for random walks on Sierpinski carpets

In the following, we present a highly efficient algorithm to iterate the ma ster equation for random walks on effectively infinite Sierpinski carpets, i.e, without surface effects. The resulting probability distribution can, f or instance, be used to get an estimate for the random walk dimension, whic h is determined by the scaling exponent of the mean square displacement ver sus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal w ith positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet need not be fixed at the beginning of the algorithm. It is restricted only by the amount of available compute r RAM. Furthermore, all the information which is needed in every step to up date the probability distribution is stored in tables. The lookup of this i nformation is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number o f time steps is decreased.