Discrete-time integrable systems and partition functions of vicious walk systems

恶性行走系统是一种多粒子系统，其中每两个粒子或行走者不能同时占据同一位置。在本研究中，我们考虑了不同类型格图上的恶性行走系统，包括正方形和三角形格。特别地，在配分函数的计算上，我们构造了格图，在格图上我们可以找到配分函数的封闭形式。为了做到这一点，我们利用具有行列式解的离散时间可积系统，如离散饥饿Toda方程。应用于超离散可积系统以及组合问题也被考虑。

Vicious walk systems are multiparticle systems in which every two particles, or walkers, cannot occupy the same site at the same time. In this research we consider vicious walk systems on lattice graphs of various types, including the square and triangular lattices. In particular, focusing on the calculation of partition functions, we construct lattice graphs on which we can find a closed form of partition functions. To do that, we utilize discrete-time integrable systems having determinant solutions such as the discrete hungry Toda equation. Applications to ultradiscrete integrable systems as well as combinatorial problems are also considered.