  We analytically derive, in the context of the replica formalism, the first
finite size corrections to the average optimal cost in the random assignment
problem for a quite generic distribution law for the costs. We show that, when
moving from a power-law distribution to a $\Gamma$ distribution, the leading
correction changes both in sign and in its scaling properties. We also examine
the behavior of the corrections when approaching a $\delta$-function
distribution. By using a numerical solution of the saddle-point equations, we
provide predictions that are confirmed by numerical simulations.
