$$\int d^n\theta\prod_{1\leq j<k\leq N}|\theta_j-\theta_k|^2
e^{-{\lambda}/2\sum_{k=1}^N\theta^2_k}=
\frac{(2\pi)^{N/2}}{{\lambda}^{N^2/2}}\biggl(\,\prod_{p=1}^Np!\biggr).$$ 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}
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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 