paths, which is given by the representation , in the limit, when the number of links $K=m+2 L$ increases. We shall assume, that the restriction $1\ll m\ll L$ is valid which means that $m$ increases moderately in the comparison with the increase of the number of turns $L$: for instance, let $L$ increase as $m^2$. Using the known asymptotical expansion of the logarithm of the gamma-function (see Appendix) [@27], one can estimate the binomial coefficient $C_K^L$. Restricting ourselves by the first order of smallness, we obtain: $$C_K^L\approx\frac{2^K}{{\sqrt{\pi L}}}e^{-m^2/(4L)}
\biggl(1-\frac{m}{2L}\biggl(1-\frac{m^2}{4L}\biggr)\biggr)\approx
\frac{2^K}{{\sqrt{\pi L}}}\biggl(1-\frac{m^2}{4L}\biggr)
\sim\frac{2^{2 L}}{{\sqrt{\pi L}\,}}.
\label{49}$$ The second approximate equality in  takes place if $L$ is increasing faster than $m^2$. The estimate  characterizes an increase of the number of the trajectories  for a single pedestrian.

The third term in  can be written as $2A(m,L)C_K^LC_K^L$, where $$A(m,L)\equiv-m+\sum_{l=1}^m\frac{(L+m+2-l)_{l-1}}{(L+1)_{l-1}}\,.
\label{50}$$ Standard notation $(\alpha)_{n}$ for the Pochhammer’s symbol is used in  [@27]. Applying again an expansion of the logarithm of the gamma-function (A1), we can estimate $A(m,L)$ : $$A(m,L)\simeq mZ_1(m,L)-Z_0(m,L)+\mathcal{O}(m^{-1}),
\label{51}$$ where $$\begin{aligned}
&Z_0(m,L)\equiv e^{m^2/4L}\biggl(1+\frac{m}{L^2}\sum_{l=0}^{m/2}
e^{-l^2/L}\biggl(\frac{m^2}{4}-l^2\biggr)\biggr),
\\
&Z_1(m,L)\equiv-1+e^{m^2/4L}\,\frac{2}{m}\sum_{l=0}^{m/2}e^{-l^2/L}.
\end{aligned}
\label{52}$$ Let the values $m$ and $L$ increase with the ratio $L/m^2$ being finite and of order of unity. It can be shown (by means of 