  In the framework of a toy model which possesses the main features of QCD in
the high energy limit, we conduct a numerical study of scattering amplitudes
constructed from parton splittings and projectile-target multiple interactions,
in a way that unitarizes the amplitudes without however explicit saturation in
the wavefunction of the incoming states. This calculation is performed in two
different ways. One of these formulations, the closest to field theory,
involves the numerical resummation of a factorially divergent series, for which
we develop appropriate numerical tools. We accurately compare the properties of
the resulting amplitudes with what would be expected if saturation were
explicitly included in the evolution of the states. We observe that the
amplitudes have similar properties in a small but finite range of rapidity in
the beginning of the evolution, as expected. Some of the features of
reaction-diffusion processes are already present in that range, even when
saturation is left out of the model.
