The procedure described can also be applied to other correlators $G_{j;l}^{ab}(t)$ , where $a,b\in\{+,-\}$.

Random walks {#sec3}
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As it has been shown in [@15], [@16], the flips of spins on a one-dimensional lattice may be associated with a random movements of walkers. Indeed, let us consider a state of the $XX$ Heisenberg chain, which corresponds to the ferromagnetic ordering of $M$ spins: $|\!\!\Uparrow\rangle\equiv\bigotimes_{n=1}^M|\!\!\uparrow\rangle_n$ (i.e., all spins are oriented “up”). Consider the average of the following type: $$F_{j;l}(\lambda)\equiv
\langle\Uparrow\!|\sigma_{j}^{+}e^{-\lambda H_0}
\sigma_{l}^{-}|\!\Uparrow\rangle,
\label{17}$$ where the notation $H_0$ implies that the zero magnetic field $h=0$ is taken in the Hamiltonian ,  (we shall omit the same subscript for the corresponding matrices ${\widehat H}^{\pm}$ ), and $\lambda\in\mathbb C$ is an “evolution” parameter. “Up” (or “down”) direction of spin corresponds to the empty (or filled) site. Differentiating ${F_{j;l}({\lambda})}$  and expanding the commutator $\lbrack H_0,\sigma_j^{+}\rbrack$ we obtain the difference–differential equation: $$\frac{d}{d\lambda}\,F_{j;l}({\lambda})=
\frac12\,(F_{j+1;l}({\lambda})+F_{j-1;l}({\lambda}))
\label{18}$$ (and similar equation can be also obtained for the fixed index $j$). Solution of the given equation is specified by the boundary conditions imposed on the lattice argument, and by the initial condition at $\lambda=0$.

The average $F_{j;l}(\lambda)$ can be considered as the generating function of the trajectories with random turns that 