  The Kuiper belt is a population of icy bodies beyond the orbit of Neptune.
The complex orbital structure of the Kuiper belt, including several categories
of objects inside and outside of resonances with Neptune, emerged as a result
of Neptune's migration into an outer planetesimal disk. An outstanding problem
with the existing migration models is that they invariably predict excessively
large resonant populations, while observations show that the non-resonant
orbits are in fact common (e.g., the main belt population is 2-4 times larger
than Plutinos in the 3:2 resonance). Here we show that this problem can
resolved if it is assumed that Neptune's migration was grainy, as expected from
scattering encounters of Neptune with massive planetesimals. The grainy
migration acts to destabilize resonant bodies with large libration amplitudes,
a fraction of which ends up on stable non-resonant orbits. Thus, the
non-resonant--to--resonant ratio obtained with the grainy migration is higher,
up to ~10 times higher for the range of parameters investigated here, than in a
model with smooth migration. The grainy migration leads to a narrower
distribution of the libration amplitudes in the 3:2 resonance. The best fit to
observations is obtained when it is assumed that the outer planetesimal disk
below 30 AU contained 1000-4000 Plutos, or ~1000 bodies twice as massive as
Pluto. We find that the probability for an outer disk object to end up on a
stable orbit in the Kuiper belt is ~10^{-3}. Together, these results are
consistent with having only two (known) Pluto-mass bodies in the Kuiper belt
today (Pluto and Eris). The combined mass of Pluto-class objects in the
original disk was ~2-8 Earth masses (M_Earth), which represents 10-40% of the
estimated disk mass (M_disk ~ 20 M_Earth).
