---
abstract: 'A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the $XX$ Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called “vicious” random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the $XX$ magnet, can be interpreted as the generating function of paths made by a single walker in a medium characterized by a non-constant number of vicious neighbors. The answers are obtained for a number of paths made by the described walker from some fixed lattice site to another sufficiently remote one. Asymptotical estimates for the number of paths are provided in the limit, when the number of steps is increased.'
author:
- |
    $${}$$\
    [**N. M. Bogoliubov$^\dagger$, C. Malyshev$^\ddagger$**]{}\
    [Steklov Mathematical Institute, St.-Petersburg Department, RAS]{}\
    [Fontanka 27, St.-Petersburg, 191023, Russia]{}\
    \[0.5cm\] $^\dagger$ e-mail: [*bogoliub@pdmi.ras.ru*]{}\
    $^\ddagger$ e-mail: [*malyshev@pdmi.ras.ru*]{}
title: |
     $${}$$\
    [**The correlation functions**]{}\
    [**of the ${\bf XX}$ Heisenberg magnet and**]{}\
    [**random walks of vicious walkers** ]{}
---

Introduction {#sec1}
============

The random walks is a classical problem both for combinatorics and statistical physics. The problem of enumeration of the paths made by the, so-called, *vicious* walkers on the one-dimensional lattice has been formulated and investigated in details by Fisher [@1]. It is supposed that any two vicious walkers, provided both arrive at the same lattice site, annihilate not only one another but all other walkers as well. The problem mentioned still continues to attract considerable attention both of physicists and mathematicians [@2]–[@9]. Closely related problems arise also in the studies of the self-organized criticality [@10], domain walls [@11], and polymers [@12]. In paper [@13] a random walks of the annihilating particles on a ring was considered. In paper [@14] a random turns walks on a semi-axes with a possible creation of the particles at the origin was studied.

It has been shown in [@15], [@16] that the correlation functions, obtained as an averages over the ferromagnetic state of the $XX$ Heisenberg chain, can be used for enumeration of the paths of random walks of vicious walkers. In the present paper the averages of special type are investigated both for the case of ferromagnetic state and for superposition of the eigen-states of 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}
=======================================

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 $S^z$ is $z$-component of the total spin operator, and $\gamma$ is the anisotropy parameter. The local spin operators $\sigma^\pm_n=(\sigma^x_n\pm i\sigma^y_n)/2$ and $\sigma^z_n$ are given by the Pauli matrices, which depend on the lattice argument $n\in\mathcal M\equiv\{1,2,\dots,M\}$, where $M=0\pmod{2}$. The corresponding commutation relations have the form: $$[\sigma^+_k,\sigma^-_l]\,=\,\delta_{k,l}\sigma^z_l\,,\quad
[\sigma^z_k,\sigma^\pm_l]\,=\,\pm2\delta_{k,l}\sigma^{\pm}_l\,.$$ The introduced *hopping matrix* $\Delta^{(s)}$ is defined by the following entries: $$\Delta^{(s)}_{nm}\equiv\frac12\,
(\delta_{|n-m|,1}+s\delta_{|n-m|,M-1}), \label{2}$$ where ${\delta}_{n,l}$ is the Kronecker symbol, and $s$ can take two values: $s=\pm$. It is assumed that the periodic boundary conditions $\sigma^{\alpha}_{n+M}=\sigma^{\alpha}_n$ are imposed for any $n\in\mathcal M$. The Hamiltonian $H$  is reduced to the Hamiltonian of the $XX$ magnet at zero value of the parameter $\gamma$.

The most general definition of the time $t$ and temperature $T\equiv
1/\beta$ dependent correlation functions of the model under consideration looks as follows: $$G_{j;l}^{ab}(t)\equiv\frac{1}Z
\Tr(\sigma^a_{j}(0)\sigma^b_l(t)e^{-\beta H}),\qquad
Z\equiv\Tr(e^{-\beta H}),
\label{3}$$ where $\sigma^b_l(t)\equiv e^{itH}\sigma^b_le^{-itH}$ and $\Tr$ means the averaging with respect to all eigen-states of the Hamiltonian $H$. In addition, the normalization involves the partition function $Z$. Calculation of the correlators  has been carried out in [@21] as averaging over all eigen-functions of the Hamiltonian of the $XX$ magnet. In [@21] the main attention has been paid to a relationship between the correlation functions and the Fredholm determinants in the thermodynamic limit. In the present paper we shall consider the $XX$ chain only and denote its Hamiltonian by $H$.

To calculate the averages  one can use a representation of the canonical Fermi variables $c_j$, $c^{\dagger}_j$, $j\in{\mathcal M}$ through the spin variables [@19], [@20]. The corresponding Jordan–Wigner transformation has the form: $$\sigma^+_n=\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr)c_n,\qquad
\sigma^-_n=c_n^{\dagger}\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr),
\qquad n\in{\mathcal M},
\label{4}$$ where $\sigma^z_j=1-2c_j^{\dagger} c_j$. The periodic boundary conditions for the spin variables lead to the following boundary conditions for the Fermi variables: $$c_{M+1}=(-1)^{\mathcal{N}}c_1,\qquad
c^{\dagger}_{M+1}=c^{\dagger}_1(-1)^{\mathcal{N}},
\label{5}$$ where $\mathcal{N}=\sum_{n=1}^Mc_n^{\dagger} c_n$ is the operator of the total number of particles. The Hamiltonian $H$  takes the following form in the fermion representation [@19], [@20] $$H=H^+P^++H^-P^-,
\label{6}$$ where $P^{+}$ ($P^{-}$) are projectors on the states characterized by an even/odd number of fermions: $$P_++P_-=\mathbb I\,,\quad P_+-P_-=(-1)^{\mathcal N}\,.$$ The operators $H^{\pm}$  are formally identical, their superscripts $s=\pm$ point out an appropriate specification of the boundary conditions : $$c_{M+1}\,=\,-s\,c_1\,,\quad c^{\dagger}_{M+1}\,=\,- s\,c^{\dagger}_1\,.$$ To put it differently, the quadratic in the fermion variables operators $H^\pm$ has the following representation: $$H^{\pm}=c^{\dagger}\widehat H^{\pm} c-\frac{Mh}2,\qquad
\widehat H^{\pm}=-\hat\Delta^{(\mp)}+h{\hat I},
\label{7}$$ where the matrices ${\widehat H}^{\pm}$ are expressed through the hopping matrices  and $\hat I$ is the unit matrix: $${\widehat H}^\pm\,=\,\left(
\begin{array}{cccccc}   h   & -1/2 &     &   &      & \pm1/2 \\
        -1/2  &  h   & -1/2  &        &         &       \\
              & -1/2 &  h    & -1/2   &         &        \\
              &      &       & \dots  &         &        \\
              &      &       &  -1/2  & h       & -1/2   \\
     \pm 1/2  &      &       &        & -1/2    &  h
\end{array}\right)$$ (only non-zero entries are displayed). Besides, the short-hand notations $c^{\dagger}$ and $c$ are used in  for the $M$-dimensional row and column with the entries $c_n^{\dagger}$, $c_n$, $n\in\mathcal M$.

In particular, the correlator  at $a=b=z$ takes the following form in the representation  [@18], [@22], [@23]: $$G_{j;l}^{zz}(t)=1-\frac{2}Z\Tr(c^{\dagger}_{j}c_{j}e^{-\beta H})-
\frac{2}Z\Tr(c^{\dagger}_lc_le^{-\beta H})+
\frac{4}Z\Tr(c^{\dagger}_{j}c_{j}
e^{itH}c^{\dagger}_lc_le^{-(\beta+it)H}).
\label{8}$$ In order to calculate , it is convenient to introduce the generating functional [@18]: $$\mathcal G\equiv\mathcal G(S,T\mid\lambda,\nu)=
\frac{1}Z\Tr(e^Se^{-\lambda H}e^Te^{-\nu H}),
\label{9}$$ where $\lambda$, $\nu$ are the complex parameters, $\lambda+\nu=\beta$. The quadratic operators $S\equiv c^{\dagger}{\widehat S}c$ and $T\equiv c^{\dagger}{\widehat T}c$, used in , are defined by means of the matrices $\widehat S=\diag\{S_1,S_2,\dots,S_M\}$ and $\widehat T=\diag\{T_1,T_2,\dots,T_M\}$. For instance, the last term in right-hand side of is obtained from  in the following way: $$\lim_{\substack{S_n,T_n\to0,\\n\in \mathcal M}}\,\lim_{\lambda\to-it}\,
\lim_{\nu\to\beta+it}\frac{\partial}{\partial S_j}\,
\frac{\partial}{\partial T_l}\,\mathcal G(S,T\mid\lambda,\nu).
\label{10}$$ The trace in right-hand side of  can be re-written by means of  [@18]: $$\Tr(e^Se^{-\lambda H}e^Te^{-\nu H})=
\frac12\,(\mathcal G^+_{\Fa}Z^+_{\Fa}+\mathcal G^-_{\Fa}Z^-_{\Fa}
+\mathcal G^+_{\Ba}Z^+_{\Ba}-\mathcal G^-_{\Ba}Z^-_{\Ba}),
\label{11}$$ where $$\begin{aligned}
&\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}),
\\
&\mathcal G^{\pm}_{\Ba}Z^{\pm}_{\Ba}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^T(-1)^{\mathcal N}e^{-\nu H^{\pm}}),
\end{aligned}
\label{12}$$ and $$Z^{\pm}_{\Fa}\,=\,\Tr(e^{-\beta H^{\pm}})\,,\quad
Z^{\pm}_{\Ba}\,=\,\Tr((-1)^{\mathcal N}e^{-\beta H^{\pm}})\,.$$ Moreover, for the partition function $Z$ we obtain the representation: $$Z=\frac12\,(Z^+_{\Fa}+Z^-_{\Fa}+Z^+_{\Ba}-Z^-_{\Ba})\,.$$ In the thermodynamic limit the terms with the subscript $\Ba$ are mutually compensated, therefore, in order to obtain $\mathcal G$  it is enough to calculate $\mathcal G^{\pm}_{\Fa}$.

The considered fermion representation is characterized by the existence of the Fock state $|0\rangle$ common for both operators $H^{+}$ and $H^{-}$, and satisfying the relations $c_k|0\rangle=0$, $k\in\mathcal M$. However, the corresponding coherent states over $|0\rangle$, $$\begin{array}{rcl}
&&\mid\!z\big\rangle\,\equiv\,\exp \Big(\sum\limits^M_{k=1} c^\dagger_k
z_k\Big) \mid\!0\big\rangle\,\equiv\,\exp (c^\dagger z)\mid\!0\big\rangle\,,\\ [0.5cm]
&&\big\langle z^*\!\mid\,\equiv\,\big\langle 0\!\mid \exp
       \Big(\sum\limits^M_{k=1} z^*_k c_k \Big)\,\equiv\,
     \big\langle 0\!\mid\exp (z^* c)\,,
\end{array}$$ are different for $H^+$ and $H^-$. Here the short-hand notations $z^*\equiv(z^*_1,\dots,z^*_M)$ and $z\equiv(z_1,\dots,z_M)$ are used for the sets of independent Grassmann parameters $z_k,z^*_k$, $k\in {\mathcal M}$ (it is appropriate to omit the extra index $\pm$ in $z^*$, $z$). Besides, $\sum_{k=1}^{M} c^{\dagger}_kz_k\equiv c^{\dagger} z$, $\prod_{k=1}^M dz_k\equiv dz$. Let us calculate $\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}$ in  using the representation of the trace in the Grassmann integration formalism [@18]: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int dz\,dz^*\,e^{z^*z}
\langle z^*|e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}|z\rangle.
\label{13}$$

In order to represent the right-hand side of this equality as the functional integral, let us introduce $L$ new copies of the coherent states $|x(I)\rangle$, $\langle x^*(I)|$, where $I\in\{1,2,\dots,L\}$. Each of the $2L$ multi-indices $x^*(I)$, $x(I)$ is expressed by $M$ independent Grassmann parameters. Using the decompositions of unity one can represent the right-hand side of  as the $(L+1)$-fold multiple integral. In order to express the quasi-periodicity condition it is convenient to introduce the auxiliary variables: $$-{\widehat E}x(0)=x(L+1)\equiv z,\qquad
-x^*(L+1)=x^*(0){\widehat E}^{-1}\equiv z^*,
\label{14}$$ where ${\widehat E}\equiv e^{\widehat S}e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}$. Tending $L$ to infinity, we obtain the functional integral over the space of the trajectories $x^*(\tau)$, $x(\tau)$, where $\tau\in {\mathbb R}$: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int e^S\,d\lambda^*\,d\lambda
\prod_{\tau} dx^*(\tau)\,dx(\tau).
\label{15}$$ The action functional $S\equiv\int L(\tau)\,d\tau$ is expressed through the Lagrangian $L(\tau)$: $$L(\tau)\equiv x^*(\tau)\biggl(\frac{d}{d\tau}-
\widehat H^{\pm}\biggr)x(\tau)+
J^*(\tau)x(\tau)+x^*(\tau)J(\tau),$$ where $$J^*(\tau)\equiv\lambda^*(\delta(\tau)\hat I+
\delta(\tau-\nu){\widehat E}^{-1}),\qquad
J(\tau)\equiv(\delta(\tau)\hat I+
\delta(\tau-\nu)\widehat E)\lambda.$$ The integration over the auxiliary Grassmann variables $\lambda^*$, $\lambda$ in  guarantees the fulfilment of the constraints . The $\delta$-functions in $J^*(\tau)$, $J(\tau)$ reduce $\tau\in\mathbb R$ to $\tau\in[0,\beta]$. The stationarity conditions $\delta S/\delta x^*=0$, $\delta S/\delta x=0$ result in the following regularized answer [@18]: $$\mathcal G^{\pm}_{\Fa}={\det}\biggl(\hat I+
\frac{e^{(\beta-\nu){\widehat H}^{\pm}}e^{\widehat S}
e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}-{\hat I}}
{\hat I+e^{\beta{\widehat H}^{\pm}}}\biggr).
\label{16}$$ Furthermore, we substitute  into  and pass to the momentum representation. The procedure described can also be applied to other correlators $G_{j;l}^{ab}(t)$ , where $a,b\in\{+,-\}$.

Random walks {#sec3}
============

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 start at the $l$-th site and end up at the $j$-th site. Indeed, let us introduce the notation $\mathcal D^K_{\lambda}$ for the operator of differentiation of $K$-th order with respect to $\lambda$ at the point $\lambda=0$. The application of $\mathcal D^K_{\lambda}$ to the average  leads to the answer: $$\mathcal D^K_{\lambda}\bigl[F_{j;l}({\lambda})\bigr]=
\langle\Uparrow|\sigma_{j}^{+}(-H_0)^K\sigma_{l}^{-}|\Uparrow\rangle=
\sum_{n_1,\dots,n_{K-1}}\Delta^{(+)}_{jn_{K-1}}\dots
\Delta^{(+)}_{n_2n_1}\Delta^{(+)}_{n_1l}.
\label{19}$$ The right hand side of  coincides with the entry at the crossing of the $j$-th row and the $l$-th column of the matrix given by the product of $K$ copies of the hopping matrix . Each matrix in this product corresponds to a transition between the two nearest sites of the lattice. After multiplication by $2^{K}$ (this is due to the accepted normalization of the matrix ), the right-hand side of  gives the number of the trajectories that consist of $K$ steps and are connecting the $l$-th and $j$-th sites. Let us denote this number by $|P_K(l\rightarrow j)|$.

Let $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ be a number of trajectories consisting of $K$ links made by $N$ vicious walkers in the random turns model. Here, the initial and final positions of the walkers on the sites are given respectively by the sequences $l_1 > l_2 > \dots > l_N$ and $j_1 > j_2 >\dots > j_N$. Let us consider the $N$-point correlation function ($N\leq M$): $$F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})=
\langle\Uparrow\!\!|\sigma_{j_1}^{+}\sigma_{j_2}^{+}\dots
\sigma_{j_N}^{+}e^{-\lambda H_0}\sigma_{l_1}^{-}
\sigma_{l_2}^{-}\dots\sigma_{l_N}^{-}|\!\!\Uparrow\rangle.
\label{20}$$ The present correlator is related to enumeration of the admissible trajectories which are traced by $N$ vicious walkers. Indeed, the application of the operator $\mathcal D^K_{\lambda/2}$ to  results in the average of the type $$\langle\Uparrow\!\!|\sigma_{j_1}^{+} \sigma_{j_2}^{+} \dots
\sigma_{j_N}^{+}(-2H_0)^K \sigma_{l_1}^{-}\sigma_{l_2}^{-} \dots
\sigma_{l_N}^{-} |\!\!\Uparrow\rangle\,.$$ This average provides the numbers $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ that can be established with the help of the commutator $$\lbrack H_0,\sigma_{l_1}^{-}\sigma_{l_2}^{-}\dots\sigma_{l_K}^{-}]=
\sum_{k=1}^K\sigma_{l_1}^{-}\dots\sigma_{l_{k-1}}^{-}
[H_0,\sigma_{l_k}^{-}]\sigma_{l_{k+1}}^{-}\dots\sigma_{l_K}^{-}
\label{21}$$ (in this case, differentiation with respect to $\lambda/2$, instead of $\lambda$, allows to take into account the normalization of the hopping matrix ). The condition of non-intersection of trajectories of the walkers is expressed by the vanishing of the correlation function  for any pair of coinciding indices $l_k$ or $j_p$.

Differentiating  with respect to $\lambda$ and applying , we obtain the equation: $$\frac{d}{d\lambda}\,F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)=
\frac12\sum_{k=1}^N\bigl(F_{j_1,
\dots,j_N;l_1,l_2,\dots,l_k+1,\dots,l_N}({\lambda})+
F_{j_1,\dots,j_N;l_1,l_2,\dots,l_k-1,\dots,l_N}({\lambda})\bigr).
\label{22}$$ Equation has been considered in [@16] for the case of periodicity with respect to the lattice argument and with the initial condition: $$F_{j_1, \dots, j_N;l_1, \dots, l_N}(0)\,=\,\prod_{m=1}^N\delta_{j_m,l_m}\,.$$ The function $F_{j_1,j_2, \dots, j_N; l_1,l_2, \dots, l_N}(\lambda)$ can be expressed 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, $\mathcal D^K_{\lambda/2} [F_{j_1, \dots, j_N;l_1, \dots, l_N}(\lambda)]$ has the sense of the number of trajectories of $N$ vicious walkers each of which has made $K$ steps. A different combinatorial interpretation of this object, however, can be proposed. Really, let us consider a representation of the multi-point correlator in the form of the determinant . Its entries $F_{j_r;l_s}(\lambda)$ in the case of an infinite chain are given by the Bessel function $I_{j_r-l_s}(\lambda)$ . The operator $\mathcal D^K_{\lambda/2}$ acts on the determinant as the differentiation of the product of $N$ functions: $$(f_1(x)f_2(x)\dots f_N(x))^{(K)}=
\sum_{n_1+n_2+\dots+n_N=K}P(n_1,n_2,\dots,n_N)
f^{(n_1)}_1f^{(n_2)}_2\dots f^{(n_N)}_N.
\label{27}$$ The notation $f^{(n)}\equiv d^nf(x)/dx^n$ is used here, and the coefficients $P(n_1,n_2,\dots,n_N)$ are the numbers of permutations with repeats: $$P(n_1,n_2,\dots,n_N)\equiv
\frac{(n_1+n_2+\dots+n_N)!}{n_1!\,n_2!\,\dots n_N!}.
\label{28}$$ Summation in  is over all non-negative values of $n_1,n_2,\dots,n_N$, provided their sum is equal to $K$.

Suppose further, that an $N$-dimensional (hyper-)cubic lattice of infinite extension is given, and each site of this lattice is labelled by a set of $N$ numbers. Let $\mathcal T_K(q_1,q_2, \dots
,q_N)$ be the number of the lattice trajectories that can be traced by some walker from the “initial” point $\textbf{\textit{O}}\equiv(0,0, \dots, 0)$ to a point $(q_1,q_2,\dots,q_N)$ in $K$ steps (by a single step the walker can move to one of the nearest sites). Let all numbers $q_k$ be non-negative, and let the inequality $K\geq q_1+q_2+\dots+q_N$ be fulfilled, which means that the steps that can compensate each other are allowed. Let us denote the number of these steps as $2L$, $$L\equiv\frac{K-q_1-q_2-\dots-q_N}{2}\,.
\label{29}$$ Taking into account , the following formula for the number of paths takes place: $$\mathcal T_K(q_1,q_2,\dots,q_N)=\sum_{L_1+L_2+\dots+L_N=L}
P(q_1+L_1,q_2+L_2,\dots,q_N+L_N,L_1,L_2,\dots,L_N),
\label{30}$$ where summation is taken over all non-negative values of $L_1,L_2,\dots,L_N$, provided that their sum is equal to $L$, and the formula  for the number of permutations with repeats is used.

Turning back to the function $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})$ let us define the matrix $(n_{rs})_{1\leq r,s\leq N}$ with the entries $n_{rs}\equiv j_r-l_s$. Then, we arrive to the following

The number of trajectories consisting of $K$ links, which are traced by $N$ vicious walkers on an axis, is expressed through the number of trajectories of the same “length” $K$, which are traced by a single walker travelling over sites of $N$-dimensional lattice of infinite extension: $$\begin{aligned}
|P_K(l_1,\dots,l_N\rightarrow j_1,\dots,j_N)|&\equiv
\mathcal D^K_{\lambda/2}\bigl[F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)\bigr]=
\\
&=
\sum_{S_{a_1,a_2,\dots,a_N}}(-1)^{\mathcal P_S}
\mathcal T_K(n_{a_11},n_{a_22},\dots,n_{a_NN}),
\end{aligned}
\label{31}$$ where summation is taken over all permutations $S_{a_1,a_2,\dots,a_N}\equiv$ $S(\begin{smallmatrix}1,&2,&\dots,&N \\ a_1,&a_2,&\dots,&a_N\end{smallmatrix})$ of the numbers $1,2,\dots,N$, and $\mathcal P_S$ implies a parity of a specific permutation.

In order to verify  one should develop the determinant  by a row or by a column and then apply the induction using the relations –.

Let us calculate, for instance,  at $N=2$: $$\mathcal T_K(q_1,q_2)=C_{q_1+q_2+2L}^{q_1+L}
\sum_{k=0}^LC^{L-k}_{q_1+L}C_{q_2+L}^{k}=C_K^{q_1+L}C_K^L,
\label{32}$$ where $L=(K-q_{1}-q_{2})/2$ denotes one-half of the total number of turns. Then, using  we obtain: $$\mathcal D^K_{\lambda/2}\bigl[F_{j_1,j_2;l_1,l_2}(\lambda)\bigr]=
\mathcal T_K(n_{11},n_{22})-\mathcal T_K(n_{21},n_{12})=
\begin{vmatrix}
C_K^L&C_K^{L+n_{21}}
\\[1mm]
C_K^L&C_K^{L+n_{11}}
\end{vmatrix},
\label{33}$$ where $L=(K-n_{11}-n_{22})/2$ and the equality $n_{11}+n_{22}=n_{12}+n_{21}$ is used.

Representation of the entries $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}(\lambda)$  in the integral form  allows to obtain the following expression [@15]: $$\begin{aligned}
F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})={}&
\frac{e^{{\lambda}N}}{N!}\prod_{i=1}^N
\biggl(\,\int_{-\pi}^{\pi}\frac{d\theta_i}{2\pi}\biggr)
e^{-{\lambda}\sum_{k=1}^N(1-\cos\theta_k)}\times{}
\\
&\times
{S}_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},\dots,
e^{i\theta_N})\prod_{1\leq j<k\leq N}
|e^{i\theta_j}-e^{i\theta_k}|^2,
\end{aligned}
\label{34}$$ where $S_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},
\dots,e^{i\theta_N})$  is the Schur function [@24], $$S_{\boldsymbol{\pi}}(x_1,x_2,\dots,x_N)\equiv
\frac{\det(x_j^{\pi_k+N-k})_{1\leq j,k\leq N}}
{\det(x_j^{N-k})_{1\leq j,k\leq N}}\,.
\label{35}$$ The Schur function depends on the partition $\boldsymbol{\pi}=(\pi_1,\pi_2,\dots,\pi_N)$ defined by a sequence of non-negative integers, which are ordered according to non-strict decreasing: $\pi_1\geq\pi_2\geq\dots\geq\pi_N\geq0$. In virtue of translational invariance it is always possible to choose the numbers $l_1$ $>$ $l_2>\dots>l_N\geq-N$ for the initial position of the walkers and to define the elements of the partition by the equalities $\pi_k=l_k+k$. In order to calculate the leading asymptotics of the generating function in the limit ${\lambda}\rightarrow\infty$, let us transform the integral  into the following integral [@7], [@25]: $$\int d^n\theta\prod_{1\leq j<k\leq N}|\theta_j-\theta_k|^2
e^{-{\lambda}/2\sum_{k=1}^N\theta^2_k}=
\frac{(2\pi)^{N/2}}{{\lambda}^{N^2/2}}\biggl(\,\prod_{p=1}^Np!\biggr).$$ It is a special case of the *Mehta integral*, which arises in the theory of the *Gaussian matrix ensembles*. Finally, we obtain the following asymptotics of the generating function for the trajectories traced by $N$ vicious walkers: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})\simeq
\mathcal A\frac{e^{{\lambda}N}}{{\lambda}^{N^2/2}}\,,\qquad
\mathcal A=\frac{\prod_{p=1}^{N-1}p!}{(2\pi)^{N/2}}
\prod_{1\leq j<k\leq N}\frac{l_j-l_k}{k-j}\,,$$ where the well known formula for ${S}_{\boldsymbol{\lambda}}(1,1,\dots,1)$ is taken into account in $\mathcal A$ [@9], [@25]. Therefore, the power-like behavior of $F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})$ is characterized by the exponent $\xi=-N^2/2$.

Random walks over superposition of the eigen-states {#sec3.2}
---------------------------------------------------

The eigen-functions of the $XX$ Hamiltonian, given by the relations , , are constructed as combinations of the states, obtained by “flipping” of $N$ spins in the state $|\!\!\Uparrow\rangle$ [@21]. Indeed, let us consider all admissible strict partitions $\boldsymbol{\mu}=(\mu_1,\mu_2, \dots, \mu_N)$, where $M\geq\mu_1>\mu_2> \dots >\mu_N\geq1$, and establish a correspondence between each partition and an appropriate sequence of zeros and unities: $\bigl\{e_k\equiv e_k(\boldsymbol{\mu})\bigr\}_{k\in\mathcal M}$, where $e_k=\delta_{k,\mu_n}$, $1\leq n\leq N$. The required eigen-function is defined as: $$|\Psi_N(u_1,\dots,u_N)\rangle=
\sum_{\{e_k(\boldsymbol{\mu})\}_{k\in\mathcal M}}
\Upsilon_N(\{u_k\}\!\mid\boldsymbol{\mu})
(\sigma_{M}^{-})^{e_{M}}(\sigma_{M-1}^{-})^{e_{M-1}}\dots
(\sigma_{1}^{-})^{e_1}|\Uparrow\rangle,
\label{36}$$ where summation is taken over all strict partitions $\boldsymbol{\mu}$ of the given type. The number of such partitions is expressed through the number of permutations with repeats : $P(N,M-N)=C_{M}^N$. The wave functions satisfy the periodic boundary conditions, $$\Upsilon_N(\{u_k\}|\boldsymbol{\mu})\equiv
\det(u_k^{2\mu_l})_{1\leq k,l\leq N}
\label{37}$$ are parametrized by the partitions $\boldsymbol{\mu}$ and by different, up to permutation, sets $\{u_1, \dots, u_N\}$ of solutions of the Bethe equations: $$u_k^{2M}=(-1)^{N-1},\qquad1\leq k\leq N.
\label{38}$$ These solutions have the form: $u_k^2=e^{i2\pi I_k/M}$, where $I_k$ are integers or half-integers (this depends on parity of $N$). Due to the antisymmetry of  with the respect to permutations of the parameters $u_k$, it is sufficient to restrict oneself to the strict partitions $M\geq I_1>I_2>\dots>I_N \geq1$ in order to guarantee the single-valuedness of $u_k$. With the help of  the corresponding normalized average $$\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N\equiv
\frac{\langle\Psi_N|\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}
|\Psi_N\rangle}{\langle\Psi_N\mid\Psi_N\rangle}
\label{39}$$ can be represented as a linear combination of $(N+1)$-point generating functions . Therefore, this average is related to the number of random walks of $N+1$ pedestrians. The initial and the final positions of one of them are fixed at $l_1=1$ and $j_1=m+1$, respectively, while for the rest (*virtual*) pedestrians the choice of their initial and the final positions is arbitrary.

Calculation of equation  is of interest in the thermodynamic limit, when $M$ and $N$ are growing (their ratio remains finite), which means that the number of virtual pedestrians is increasing. In this limit [@26] $$\widetilde F_{m+1;1}({\lambda})\equiv
\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N
\bigr|_{M,N\gg1}\stackrel{\defa}{=}\Tr^{\prime}(\sigma_{m+1}^{+}
e^{-{\lambda}H_0}\sigma_1^{-}),
\label{40}$$ where the notation $\Tr^{\prime}$ points out that the procedure presented in Section \[sec2\] is used for the calculation of the normalized average. The difference-differential relation, analogous to the equation  is valid for ${\widetilde F}_{m+1;1}({\lambda})$ : $$\begin{aligned}
&\frac d{d{\lambda}}\,{\widetilde F}_{m+1;1}({\lambda})=
\frac12\,({\widetilde F}_{m;1}({\lambda})+
{\widetilde F}_{m+2;1}({\lambda}))-
\Tr^{{\prime}}(H_0\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-})-{}\notag
\\
&\qquad{}-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr)-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m+2}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr).
\label{41}\end{aligned}$$ The form of the present equation makes it possible to suppose that the average ${\widetilde F}_{m+1;1}({\lambda})$ can also be of interest as a generating function of the random walks.

Let us turn to the calculation of ${\widetilde F}_{m+1;1}({\lambda})$ in the fermionic representation . It is convenient to reduce the problem to calculation of the generating function of the form: $$\widetilde{\mathcal G}\equiv\Tr^{\prime}
(e^Sc_{m+1}e^{-\lambda H_{0}}c^{\dagger}_1e^{-\nu H_{0}}),
\label{42}$$ where the operator $S$ is defined just like in  (i.e., by means of the matrix $\widehat S = \break \diag\{S_1,S_2,\dots,S_M\}$). Indeed, the functional ${\widetilde F}_{m+1;1}({\lambda})$ corresponds to the choice of $\nu=0$ and $S_k=-i\pi\theta(m-k)$, where $\theta(m-k)$ is the Heavyside function, $\theta(0)=1$. The second term in the right-hand side of  corresponds to the differentiation by $\nu$ at the point $\nu=0$, in the third term we put $\nu=0$ and differentiate with respect to $S_{m+1}$. In both cases we put $S_k=-i\pi\theta(m-k)$. Taking into account the fact, that the contribution of the terms labelled by the index $\Ba$ in  is negligible at sufficiently large $M$ and $N$, we approximately obtain: $$\begin{aligned}
&\widetilde{\mathcal G}\approx
\biggl[\tr(e^{-{\lambda}{\widehat H}^{0}}{\hat e}_{1,m+1})-
\frac d{d\alpha}\biggr]\det({\hat I}+\widehat{\mathcal M}_1+
\alpha\widehat{\mathcal M}_2)\bigr|_{{\alpha}=0},
\\
&\widehat{\mathcal M}_1+\alpha\widehat{\mathcal M}_2\equiv
e^{-\nu\widehat H^{0}}e^{\widehat S}e^{-\lambda{\widehat H}^{0}}
({\hat I}+\alpha\hat e_{1,m+1}e^{-\lambda{\widehat H}^{0}}),
\end{aligned}
\label{43}$$ where $\hat{ e}_{1,m+1}\equiv
({\delta}_{1,n}{\delta}_{m+1,l})_{1\leq n,l\leq M}$. The matrix $\widehat H^{0}$ is used instead of $\widehat H^{\pm}$ since $s$ can be replaced by zero for the sufficiently large $M$.

The relation  is written in the coordinate representation. In order to pass to the momentum representation it is convenient to use certain formulas provided in [@22]. Keeping the matrix notations as in , we obtain the answer for ${\widetilde F}_{m+1;1}({\lambda})$ (in the limit $M\to\infty$, the corresponding operations should be understood in the sense of the operations over the corresponding integral operators [@26]): $${\widetilde F}_{m+1;1}({\lambda})=
\det({\hat I}+\widehat{\mathcal U}_m)
\biggl[\tr(e^{-{\lambda}\hat{\varepsilon}_{0}}\breve e_{1,m+1})-
\tr\biggl(\frac{\widehat{\mathcal V}_{m}}{{\hat I}+
\widehat{\mathcal U}_m}\biggr)\biggr]
\label{44}$$ (the notation $\tr$, for instance, corresponds to the trace of $M\times M$ matrices). The matrices $\widehat{\mathcal U}_m$, $\widehat{\mathcal V}_{m}$, ${\breve e}_{1,m+1}$ are given by the corresponding momentum representations of the matrices $\widehat{\mathcal M}_1$, $\widehat{\mathcal M}_2$, ${\hat
e}_{1,m+1}$ . However, we shall need explicit expressions only for the traces $\tr\widehat{\mathcal U}_m$ and $\tr\widehat{\mathcal V}_{m}$ (see below). In the momentum representation, $\hat{\varepsilon}_{0}$ is a diagonal matrix of the eigen-energies of the $XX$ model at $h=0$ [@21]. Formally expanding ${\widetilde F}_{m+1;1}({\lambda})$ in the powers of $\widehat{\mathcal U}_m$ we shall obtain the answer in two lowest orders: $$\begin{aligned}
&{\widetilde F}_{m+1;1}({\lambda})\approx
F_{m+1;1}({\lambda})+F_{m+1;1}({\lambda})
\tr\widehat{\mathcal U}_m-\tr\widehat{\mathcal V}_{m},
\nonumber
\\
&\tr\widehat{\mathcal U}_m=(M-2m)F_{1;1}({\lambda}),
\label{45}
\\
&\tr\widehat{\mathcal V}_{m}=F_{m+1;1}(2{\lambda})-
2\sum_{l=1}^{m}F_{m+1;l}({\lambda})F_{l;1}({\lambda}),
\nonumber\end{aligned}$$ where the notation $F_{j;l}({\lambda})$ implies the relations . Although $M$ and $m$ are chosen to be large enough, the ratio $m/M$ is assumed to be finite. Equation  is fulfilled in each order separately by the terms presented in ${\widetilde F}_{m+1;1}$ .

By an analogy with the ferromagnetic case, let us act on ${\widetilde F}_{m+1,1}({\lambda})$  by the operator $\mathcal D^K_{\lambda/2}$. Then, in the first order we shall obtain the relation . In the second order, the answer is of the following form: $$MC_K^LC_K^L-C_K^L\sum_{l=0}^KC_K^l+2\sum_{l=1}^m
\begin{vmatrix}
C_K^{L+l-1}&C_K^L
\\
C_K^L&C_K^L
\end{vmatrix}.
\label{46}$$ By virtue of , the result of the application of $\mathcal D^K_{\lambda/2}$ to the second order function $F_{j_1,j_2;l_1,l_2}(\lambda)$ is connected, as a particular case of , with the number of the two-dimensional paths $\mathcal T_K$, and is expressed through the corresponding determinant. It means that it will be appropriate 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 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 numerical check as well), that the coefficient functions $Z_0(m,L)$ and $Z_1(m,L)$ remain finite in this case, and the contribution of $Z_0(m,L)$ is negligible in comparison with $m Z_1(m,L)$ in . One can use Eqs.  and  in order to estimate  in the leading approximation: $$\frac{2^{4L}}{\pi L}\,e^{-m^2/2L}(M+2mZ_1(m,L)-
(\pi L)^{1/2}e^{m^2/4L}).
\label{53}$$ Because of the behavior of the coefficient $Z_1(m,L)$, the corresponding contribution in  may turn out to be comparable with $M$. The relation  demonstrates that the description of the random walks considered in the representation of the superposition of the eigen-states is more complicated than the one in the ferromagnetic case. This description can be regarded as a simultaneous walks of the initial (i.e., principal) and virtual pedestrians. The ending points of the trajectories belonging to all the three segments of the dashed broken line on the figure (see the representation of two-dimensional random walks ) correspond to comparable contributions into the estimate . In certain cases, characterized by the limiting behavior of the ratio $m^2/L$, the contribution of the segment between the points $(m,0)$ and $(0,m)$ can become dominating.

Conclusion {#sec4}
==========

It is shown that the correlation functions of the $XX$ Heisenberg magnet, calculated over the superposition of the eigen-states, as well as over the ferromagnetic state, are connected with enumeration of the trajectories made by the walkers moving on the lattice. A relationship is established between the number of trajectories made by a several vicious walkers and the number of paths made by a single random turns walker on a lattice of a dimension equal to the number of the vicious walkers. Differentiation of the generating function, calculated over the superposition of the eigen-states, demonstrates a more complicated combinatorial picture than that of the ferromagnetic case. In particular, the set of the paths made by a single pedestrian is replaced by the set of trajectories made simultaneously by the principal and virtual (both vicious) pedestrians. An estimate is obtained for the number of trajectories made both by the principal and the virtual pedestrians.

Acknowledgement {#acknowledgement .unnumbered}
===============

This paper was partially supported by the Russian Foundation for Basic Research, No. 07-01-00358, and by the Russian Academy of Sciences program ,,Mathematical Methods in Non-Linear Dynamics”.

Appendix {#appendix .unnumbered}
========

Asymptotic expansion for the logarithm of the gamma-function at large $|z|$ and $|\arg z |< \pi$ has the form [@27]: $$\begin{array}{rcl}
\log\Gamma
(z\,+\,\alpha)&=&\displaystyle{\Bigl(z\,+\,\alpha\,-\,\frac12\Bigr) \log
z\,-\,z\,+\,\frac12\,\log(2\pi)}\nonumber\\[0.4cm]
&+&\displaystyle{\sum\limits_{p=1}^n
(-1)^{p+1}\,\frac{B_{p+1}(\alpha)}{p(p+1)}\,z^{-p}\,+\,
\mathcal{O}\Bigl(\frac1{z^{n+1}}\Bigr)}\,,
\end{array}\eqno(A1)$$ where $n= 1, 2, 3, \dots$. The Bernoulli polynomials $B_{n}(\alpha)$ ($A1$) are defined as follows: $$B_n(\alpha)\,=\,\sum\limits_{l=0}^n C_n^l\,B_l\,\alpha^{n-l}\,,$$ where $C_n^l$ are the binomial coefficients, and $B_l$ are the Bernoulli numbers. The first Bernoulli polynomials $B_{p}(\alpha)$ look as follows: $$\begin{aligned}
B_0(\alpha)\,=\,1\,,\qquad B_1(\alpha)\,=\,\alpha\,-\,\frac12\,,\qquad
B_2(\alpha)\,=\,\alpha^2\,-\,\alpha\,+\,\frac16\,,\nonumber\\[0.4cm]
B_3(\alpha)\,=\,\alpha^3\,-\,\frac32\,\alpha^2\,+\,\frac12\,\alpha\,,\qquad
B_4(\alpha)\,=\,\alpha^4\,-\,2 \alpha^3\,+\,\alpha^2\,-\,\frac1{30}\,,
\nonumber\end{aligned}$$ where the Bernoulli numbers $B_l$ are used: $$\begin{aligned}
B_0\,=\,1\,,\qquad B_1\,=\,-\,\frac12\,,\qquad
B_2\,=\,\frac16\,,\qquad B_4\,=\,-\,\frac1{30}\,.
\nonumber\end{aligned}$$ 0.5cm

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