Reunion probabilities of $N$ onedimensional random walkers with mixed boundary conditions
Abstract
In this work we extend the results of the reunion probability of $N$ onedimensional 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 LiebLiniger gas (LLg) of $N$ onedimensional particles. We use Bethe ansatz and Gaudin's conjecture to obtain the normalized wavefunctions and use this information to construct the propagator. As it is wellknown, 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 quasimomenta $k$ obeying the Bethe equations for that particular boundary condition.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 DOI:
 10.48550/arXiv.1311.0654
 arXiv:
 arXiv:1311.0654
 Bibcode:
 2013arXiv1311.0654P
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematical Physics