[@Levin1; @Levin2] for recent progress), but the result looks always complicated in QCD and thus difficult to play with and to interpret. Numerical studies were conducted by Salam [@S1; @S_MC; @MS], but at that time, there was no good theoretical understanding of the properties that scattering amplitudes should exhibit at high energy. In the light of our present knowledge of saturation, that enables us to characterize saturation by analyzing the properties of some traveling waves, we evaluate numerically, on toy models, how good this procedure is when only splittings and multiple exchanges are allowed. Another important highlight of the present work is that we are able to resum numerically the asymptotic series that can be constructed out of an expansion in the number of rescatterings, and which has the structure of the proper field-theoretical series of successive Pomeron exchanges.

Our study assumes the following standard picture of scattering in the QCD parton model (see Fig. \[fig:scat\]): An asymptotic hadron, made of valence partons, evolves into a set of quarks and gluons spread in impact parameter space, when its rapidity is increased. The rules of splitting (and recombination when nonlinear effects are taken into account in the evolution) of the partons 