  We study a one-dimensional model for heavy particles in a compressible fluid.
The fluid-velocity field is modelled by a persistent Gaussian random function,
and the particles are assumed to be weakly inertial. Since one-dimensional
fluid-velocity fields are always compressible, the model exhibits spatial
trapping regions where particles tend to accumulate. We determine the
statistics of fluid-velocity gradients in the vicinity of these traps and show
how this allows to determine the spatial Lyapunov exponent and the rate of
caustic formation. We compare our analytical results with numerical simulations
of the model and explore the limits of validity of the theory. Finally, we
discuss implications for higher-dimensional systems.
