not done so: We shall check boost invariance later on, but only numerically).

\[sec3C\]Expected properties of the $S$-matrix
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If the partonic evolution is linear (i.e. without saturation effects), and if the scattering takes place in the lab frame, then $S=S_\text{BK}$, and $S$ exhibits the properties listed in Sec. \[sec2B\].

We also know that, whatever the saturation mechanism is, as soon as the growth of the number of partons on each site is limited once it approaches $1/\alpha^2$, then the traveling wave solution for $S$ is modified [@BD; @MS; @BDMM; @Panja]. Essentially, its asymptotic velocity is corrected as follows [@BD; @BDMM]: $$V=V_0-\frac{\pi^2\gamma_0\chi^{\prime\prime}(\gamma_0)}
{2 (\ln(1/\alpha^2)+3\ln\ln(1/\alpha^2))^2}+\cdots
\label{velocity}$$ This equation contains the first term of an asymptotic expansion when $\ln(1/\alpha^2)\gg 1$ (It is actually valid up to ${\cal O}(\ln\ln (1/\alpha^2)/\ln^3(1/\alpha^2)$). We do not intend to go to extremely small values of $\alpha$, thus the exact value of these asymptotics will not be probed. What is interesting and universal however is that $V$ is less than $V_0$.

We will be able to extract more information from our Monte-Carlo, that we can compare with non-trivial and characteristic predictions made for reaction-diffusion systems. In particular, since the whole scattering process is stochastic, there is a dispersion 