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Runway growth is an important stage in planet formation during which largeprotoplanets form, while most of the initial mass remains in smallplanetesimals. The amount of mass converted into large protoplanets and theirresulting size distribution are not well understood. Here, we use analyticwork, that we confirm by coagulation simulations, to describe runaway growthand the corresponding evolution of the velocity dispersion. We find thatrunaway growth proceeds as follows: Initially all the mass resides in smallplanetesimals, with mass surface density \sigma, and large protoplanets startto form by accreting small planetesimals. This growth continues until growth bymerging large protoplanets becomes comparable to growth by planetesimalaccretion. This condition sets in when \Sigma/\sigma ~\alpha^{3/4} ~ 10^{-3},where \Sigma is the mass surface density in protoplanets in a given logarithmicmass 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 velocitydispersion become self-similar and \Sigma remains roughly constant, since anincrease in \Sigma by accretion of small planetesimals is balanced by adecrease due to merging with large protoplanets. We show that this growth leadsto a protoplanet size distribution given by N(>R) \propto R^{-3} where N(>R) isthe number of objects with radii greater than R (i.e., a differential power-lawindex of 4). We apply our results to the Kuiper Belt, which is a relic ofrunaway growth. Our results successfully match the observed Kuiper belt sizedistribution, they illuminate the physical processes that shaped it and explainthe total mass that is present in large Kuiper belt objects (KBOs) today. Thiswork suggests that the current mass in large KBOs is primordial and that it hasnot been significantly depleted. Abridged

Runaway Growth During Planet Formation: Explaining the Size Distribution of Large Kuiper Belt Objects