comments are true for larger values of $1/\alpha^2$, see Fig. \[fig:v2000\_1\] and \[fig:v200000\_1\]. It is useful to compute $y_F$ and $y_R$ also in that case: $$\begin{split}
y_F\simeq 18.7,\ y_R\simeq 17.0\ \ &\text{for}\ 1/\alpha^2=2\times 10^3,\\
y_F\simeq 30.1,\ y_R\simeq 43.9 \ \ &\text{for}\ 1/\alpha^2=2\times 10^5.
\end{split}
\label{valuesyFyR}$$ We check that a non-zero velocity appears for $y>y_F$. Actually, $y_F$ given in (\[valuesyFyR\]) systematically underestimates the actual value of the rapidity at which a front is formed.

So far, we see in Fig. \[fig:v20\_1\],\[fig:v2000\_1\],\[fig:v200000\_1\] that the characteristics of the front agree in the different models formulated in the center-of-mass frame only in a very small interval of rapidity, roughly consistent with the theoretical estimate $y<2y_F$. Except for large values of $1/\alpha^2$, the front velocity in the boost-invariant scheme is not reached by bare multiple scatterings in the center-of-mass frame.

It is instructive to also study the variance of the position of the front in the different variants of the model: We analyze this quantity in the next paragraph.

Variance of the front position
------------------------------

We now turn to commenting on the variance of the position of the front. This is shown in Figs. \[fig:v20\_2\],\[fig:v2000\_2\],\[fig:v200000\_2\] for different values of $1/\alpha^2$.

First, we notice a sharp 