the $XX$ magnet in zero magnetic field. The averages in question play a role of the generating functions of number of paths of the vicious walkers. The calculation of the correlation functions is carried out by means of the functional integration [@17], [@18]. The answers are obtained for the number of paths of a single pedestrian which is travelling from one chosen site to another sufficiently remote lattice site. The asymptotical estimates are obtained for the number of paths in the limit, when the number of steps (and, correspondingly, the number of random turns) is increasing.

The paper is organized as follows. Section \[sec1\] has an introductory character. The Hamiltonian of the model and general calculation of the correlation functions are discussed in Section \[sec2\]. Section \[sec3\] deals with the specific calculations and the corresponding asymptotic estimates. Discussion in Section \[sec4\] concludes the paper.

The model and the correlation functions {#sec2}
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The $XX$ magnet we are interested in is a particular limit of a more general spin model known as the $XY$ Heisenberg chain, with the Hamiltonian in the transverse magnetic field $h>0$ given by [@19], [@20]: $$\begin{aligned}
&H=H_0+\gamma H_1-hS^z,\\
&H_0\equiv-\sum^{M}_{n,m=1}\Delta^{(+)}_{nm}\sigma^+_{n}\sigma^-_{m},
\\
&H_1\equiv-\frac12\sum_{n,m=1}^M\Delta^{(+)}_{nm}
(\sigma^+_n\sigma^+_{m}+\sigma^-_n\sigma^-_{m}),\qquad
S^z\equiv\frac{1}2\sum_{n=1}^M\sigma^z_n,
\end{aligned}
\label{1}$$ where $S^z$ 