It is very important to keep the dynamics in the transverse space if one wants the model to be representative of some of the physics of QCD. However, we do not aim at incorporating all aspects of QCD: We need a simple model, that is easy to formulate in a way which can be implemented in the form of a Monte-Carlo event generator. Let us assume that there be only one type of object in the theory, which could be gluons or, more accurately, color dipoles, and that the latter may be fully characterized by a single “space” variable, which represents for example their size in the transverse plane. There is in addition an evolution variable: the rapidity. So far, we have in fact just idealized the color dipole model [@M1].

To further simplify the model, we discretize it both in space and in rapidity. The evolution rule is the following: When the rapidity $y$ is increased by one unit, a particle at position $i$ on the lattice may be replaced, with probability $\frac12$, by two offspring at respective positions $i+j$ and $i-j$. For the distribution of $j$, we choose: $$\text{Proba}(j)=\left(1-\frac{1}{e}\right)e^{-j}.
\label{probaj}$$ This rule obviously leads to an exponential increase 