  In this work we extend the results of the reunion probability of $N$
one-dimensional random walkers to include mixed boundary conditions between
their trajectories. The level of the mixture is controlled by a parameter $c$,
which can be varied from $c=0$ (independent walkers) to $c\to\infty$ (vicious
walkers). The expressions are derived by using Quantum Mechanics formalism
(QMf) which allows us to map this problem into a Lieb-Liniger gas (LLg) of $N$
one-dimensional particles. We use Bethe ansatz and Gaudin's conjecture to
obtain the normalized wave-functions and use this information to construct the
propagator. As it is well-known, depending on the boundary conditions imposed
at the endpoints of a line segment, the statistics of the maximum heights of
the reunited trajectories have some connections with different ensembles in
Random Matrix Theory (RMT). Here we seek to extend those results and consider
four models: absorbing, periodic, reflecting, and mixed. In all four cases, the
probability that the maximum height is less or equal than $L$ takes the form
$F_N(L)=A_N\sum_{k\in\Omega_{B}}\int Dz
e^{-\sum_{j=1}^Nk_j^2+G_N(k)-\sum_{j,\ell=1}^N
z_jV_{j\ell}(k)\overline{z}_\ell}$, where $A_N$ is a normalization constant,
$G_N(k)$ and $V_{j\ell}(k)$ depend on the type of boundary condition, and
$\Omega_{B}$ is the solution set of quasi-momenta $k$ obeying the Bethe
equations for that particular boundary condition.
