  Runway growth is an important stage in planet formation during which large
protoplanets form, while most of the initial mass remains in small
planetesimals. The amount of mass converted into large protoplanets and their
resulting size distribution are not well understood. Here, we use analytic
work, that we confirm by coagulation simulations, to describe runaway growth
and the corresponding evolution of the velocity dispersion. We find that
runaway growth proceeds as follows: Initially all the mass resides in small
planetesimals, with mass surface density \sigma, and large protoplanets start
to form by accreting small planetesimals. This growth continues until growth by
merging large protoplanets becomes comparable to growth by planetesimal
accretion. This condition sets in when \Sigma/\sigma ~\alpha^{3/4} ~ 10^{-3},
where \Sigma is the mass surface density in protoplanets in a given logarithmic
mass interval and \alpha is the ratio of the size of a body to its Hill radius.
From then on, protoplanetary growth and the evolution of the velocity
dispersion become self-similar and \Sigma remains roughly constant, since an
increase in \Sigma by accretion of small planetesimals is balanced by a
decrease due to merging with large protoplanets. We show that this growth leads
to a protoplanet size distribution given by N(>R) \propto R^{-3} where N(>R) is
the number of objects with radii greater than R (i.e., a differential power-law
index of 4). We apply our results to the Kuiper Belt, which is a relic of
runaway growth. Our results successfully match the observed Kuiper belt size
distribution, they illuminate the physical processes that shaped it and explain
the total mass that is present in large Kuiper belt objects (KBOs) today. This
work suggests that the current mass in large KBOs is primordial and that it has
not been significantly depleted. Abridged
