Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical properties similar to random walks. In this study, we investigate a generalized version of the reachability problem known as ARRIVAL in Path Multigraphs, which involves predicting the number of particles that will reach designated target vertices. We show that this problem is in NP and co-NP in the general case. However, we exhibit algebraic invariants for Path Multigraphs that allow us to solve the problem efficiently, even for an exponential configuration of particles. These invariants are based on harmonic functions and are connected to the decomposition of integers in rational bases.

翻译：转子游走是一种细胞自动机，通过简单的局部规则在定向多重图中确定粒子的确定性遍历路径，却能产生复杂行为。此外，这些轨迹展现出与随机游走相似的统计特性。本研究探讨了路径多重图中名为ARRIVAL的可达性问题的广义版本，该问题涉及预测到达指定目标顶点的粒子数量。我们证明该问题在一般情形下属于NP与co-NP类。然而，我们揭示了路径多重图的代数不变量，即使面对指数级粒子配置，也能高效求解该问题。这些不变量基于调和函数，并与整数在有理数基下的分解相关联。