asymptotics is approached as[^1] $$V(y)=V_0-\frac{3}{2\gamma_0}\frac{1}{y}
+\frac{3}{2\gamma_0^2}\sqrt{\frac{2\pi}
{\chi^{\prime\prime}(\gamma_0)}}\frac{1}{y^{3/2}}+\cdots
\label{Vy}$$ where the dots stand for subleading terms whose analytical expression is not yet known.

Starting from an initial condition of the form $$S(y=0,i)=1+(e^{-\alpha^2}-1)\delta_{0,i},$$ that is to say, $$\begin{split}
&P(y=0,n_{k\ne 0}=0\ \text{and}\ n_{k=0}=1)=1,\\
&P(y=0,\text{all other config.})=0,
\end{split}$$ a front is formed (i.e. blackness is reached; $S(y,0)\ll 1$) after a rapidity of the order of $$y_F=\frac{1}{\chi(0)}\ln \frac{1}{\alpha^2},
\label{yF}$$ and it relaxes to its asymptotic shape up to a resolution[^2] $\alpha^2$ after an evolution over $$y_R=\frac{1}{2\gamma_0^2\chi^{\prime\prime}(\gamma_0)}
\ln^2\frac{1}{\alpha^2}
\label{yR}$$ units of rapidity.

\[sec3\] Unitary scattering amplitudes
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\[sec3A\]Formulation of scattering
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We consider the Fock state of a particle initially at position 0 after a rapidity evolution $y$. Through the evolution, one eventually gets a system of $\{n_j\}$ particles. The probability that this system does not interact with a target consisting in a single particle at position $i$ simply reads $$e^{-\alpha^2 n_i}.$$ This, of course, is also what we shall call the $S$-matrix element for forward elastic scattering. With a target that consists in a set of $\{m_j\}$ particles on the sites indexed by $j$, the probability that there be no interaction reads $$\prod_j e^{-\alpha^2 n_j m_j}.$$ We have just assumed the complete independence of the 