Fractal dimensions and trajectory crossings in correlated random walks

We study spatial clustering in a discrete, 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 $q=2,3,\ldots$. The correlation dimension $D_2$ turns out to be a non-analytic function of the inertia parameter in this model. We calculate $D_2$ for small $\alpha$ up to the next-to-leading order in the non-analytic term.