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, $\ensuremath{\alpha}$, 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 ${D}_{q}={D}_{2}/(q\ensuremath{-}1)$ 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 $\ensuremath{\alpha}$ up to the next-to-leading order in the nonanalytic term.