  The statistics of velocity differences between very heavy inertial particles
suspended in an incompressible turbulent flow is found to be extremely
intermittent. When particles are separated by distances within the viscous
subrange, the competition between quiet regular regions and multi-valued
caustics leads to a quasi bi-fractal behavior of the particle velocity
structure functions, with high-order moments bringing the statistical signature
of caustics. Contrastingly, for particles separated by inertial-range
distances, the velocity-difference statistics is characterized in terms of a
local H\"{o}lder exponent, which is a function of the scale-dependent particle
Stokes number only. Results are supported by high-resolution direct numerical
simulations. It is argued that these findings might have implications in the
early stage of rain droplets formation in warm clouds.
