of the number of particles on sites 1 and 2 is obtained by taking two derivatives: $$\left.\frac{\partial Z}{\partial x_1 \partial x_2}
\right|_{x_k=1}=\langle n_1 n_2\rangle.$$ The generating function $Z$ obeys the evolution equation $$\begin{gathered}
Z(y+1,\{x_k\})=\frac12 Z(y,\{x_k\})\\
+\frac12\left(1-\frac{1}{e}\right)
\sum_{j=0}^\infty e^{-j} Z(y,\{x_{k-j}\})Z(y,\{x_{k+j}\}).
\label{generating}\end{gathered}$$ This equation is easily derived by considering the very first step in rapidity and with the help of the probability distribution (\[probaj\]). This is actually a version of what we call in QCD the Balitsky-Kovchegov (BK) equation [@Balitsky:1995ub; @K1; @K2] associated to this simplified model. The physical $S$-matrix element in this framework, $S_\text{BK}(y,i)$, is a particular value of $Z$, obtained from the above equation by setting $x_{k\ne i}=1$ and $x_i=e^{-\alpha^2}$. The nonlinearity in Eq. (\[generating\]) makes sure that $S$ be unitary. We will define the scattering amplitudes properly only in Sec. \[sec3\], but before, we may summarize the main known properties of $S_\text{BK}$ seen as a solution of Eq. (\[generating\]).

\[sec2B\]Basic properties of $S_\text{BK}$
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We know that $S_\text{BK}$ has the form of a wave front that travels towards larger values of $|i|$ under rapidity evolution. Its known properties are very well documented, and we refer the reader to the original papers in QCD [@MP1; @MP2; @MP3] (see also Ref. 