Random Walks with Self-similar Clusters

We construct a random walk on a lattice having a hierarchy of self-similar clusters built into the distribution function of allowed jumps. The random walk is a discrete analog of a Lévy flight and coincides with the Lévy flight in the continum limit. The Fourier transform of the jump distribution function is the continuous nondifferentiable function of Weierstrass. We show that, for cluster formation, it is necessary that the mean-squared displacement per jump be infinite and that the random walk be transient. We interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker. The difference in dimensions is related to the fractal (Hausdorff-Besicovitch) dimension of the self-similar clusters.