as the determinant of the matrix consisting of the averages of the type of  [@16]: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})=
\det\bigl(F_{j_r;l_s}({\lambda})\bigr)_{1\leq r,s\leq N}.
\label{23}$$

Random walks on the axis {#sec3.1}
------------------------

Let us consider an infinite chain ($M\to\infty$). Then, the modified Bessel function $I_{j-l}(\lambda)$ turns out to be a solution of equation , which respects the condition $F_{j;l}({0})={\delta}_{j,l}$ [@15]: $$F_{j;l}({\lambda})=I_{j-l}({\lambda})=\frac1{2\pi
}\int_{-\pi}^{\pi}d\theta\,e^{{\lambda}\cos\theta}e^{i(j-l)\theta}.
\label{24}$$ There exists the following expansion into the power series for $I_{j-l}(\lambda)$: $$I_{j-l}({\lambda})=\sum_{Q\geq|l-j|}
\frac{1}{\bigl(\frac{Q-j+l}2\bigr)!\,\bigl(\frac{Q+j-l}2\bigr)!}
\biggl(\frac{{\lambda}}2\biggr)^Q,
\label{25}$$ where the summation index $Q$ is subjected to the requirement: $Q+|j-l|=0\pmod{2}$. In the limit of large “time” (${\lambda}\rightarrow\infty$) and for moderate values of $m\equiv|l-j|$, using the known asymptotics for the Bessel function, we obtain for the generating function: $$F_{j;l}({\lambda})\simeq\frac{e^{{\lambda}}}{{\sqrt{2\pi{\lambda}}}}
\biggl(1-\frac{4m^2-1}{8{\lambda}}+\dotsb\biggr),$$ where the decay is governed by the critical exponent $\xi=-1/2$.

Let the number $K$ satisfies the relations $K\geq|l-j|$ and ${K+|j-l|=0\nobreak\pmod{2}}$. Then, differentiation of the series  leads to the binomial relation ${|P_K(l\rightarrow j)|}=C_K^L$ for the number of all lattice paths of the “length” $K$ between two sites on the infinite axis: $$|P_K(l\rightarrow j)|\equiv
\mathcal D^K_{\lambda/2}[F_{j;l}({\lambda})]=
\frac{(m+2L)!}{L!\,(m+L)!}\,.
\label{26}$$ Here $L$ denotes the one-half of the total number of turns: $L\equiv(K-m)/2$.

Let us consider now the multi-point correlation function $F_{j_1,j_2, \dots, j_N;l_1,l_2, \dots, l_N}({\lambda})$. As it has been shown above, 