to express  in the following equivalent form: $$\begin{aligned}
&(M-K)|P_{K}(l\rightarrow l+m)|^2+{}
\notag \\
&\qquad{}+
\mathcal D^K_{\lambda/2}\begin{bmatrix}
2\displaystyle\sum_{l=1}^m
\begin{vmatrix}
F_{m+1;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;1}({\lambda})
\end{vmatrix}
-\displaystyle\sum_{l=0}^K
\begin{vmatrix}
F_{m+L;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;L}({\lambda})
\end{vmatrix}
\end{bmatrix}.
\label{47}\end{aligned}$$ In other words, the result of application of $\mathcal D^K_{\lambda/2}$ to  in the second order can be reformulated in terms of the random walks of the two pedestrians (see  and ) and the squared number of walks of a single pedestrian. The summation by the index $l$ in $\tr\widehat{\mathcal V}_{m}$  can be interpreted as the summation over positions of a virtual walker in .

Using the equation , one can represent  in terms of the number of trajectories on a two-dimensional lattice: $$(M-2(m+1))\mathcal T_K(m,0)+
\sum_{l=0}^m\mathcal T_K(m-l,l)-\sum_{l=1}^L\mathcal T_K(m+l,l)-
\sum_{l=1}^L\mathcal T_K(l,m+l).
\label{48}$$ In this relation various lattice trajectories of $K$ links are enumerated. All these trajectories start at the same point $\textbf{\textit{O}}=(0,0)$ while they terminate on the segments of the dashed broken line which connects the points $(L,L+m)$, $(0,m)$, $(m,0)$, and $(L+m,L)$ (see figure). Formally, the sign of the sum is not definite though its asymptotics is positive, in general. An analogous description is expected in the higher orders as well.

![image](bog01)

Typical configuration.

Let us estimate the behavior of the number of 