Fractal dimensions and trajectory crossings in correlated random walks

We study spatial clustering in a discrete-time, one-dimensional, stochastic, toy model of heavy particles in turbulence and calculate the spectrum of multifractal dimensions D-q as functions of a dimensionless parameter, alpha, that plays the role of an inertia parameter. Using the fact that it suffices to consider the linearized dynamics of the model at small separations, we find that D-q = D-2/(q - 1) for q = 2, 3, . . . . The correlation dimension D-2 turns out to be a nonanalytic function of the inertia parameter in this model. We calculate D-2 for small alpha up to the next-to-leading order in the nonanalytic term.