超图上零化随机游动（和熄灯对策）的模推广的可达性和常返性
Reachability and recurrence in a modular generalization of annihilating random walks (and lights-out games) on hypergraphs
摘要：
我们研究一个动力系统的动机，我们以前的工作的统计物理学的社会平衡图，可以被看作是一个概括的零化行走沿两个方向：第一，相互作用的拓扑结构是一个超图;第二，"粒子数"在超图的顶点是一个元素的有限域${\bf Z}_{p}$的整数模$p$，$p\geq 3$。等价地，粒子在超图上移动，顶点上的移动粒子被给定超边中每个相邻点上的一个不可区分的副本所取代;当顶点上的粒子数量达到$p$时，它们集体湮灭。我们所研究的系统也可以看作是某些熄灯游戏到有限域和超图拓扑的自然推广。我们的结果表明，在相互作用超图性质的充分条件下，存在一个判定该动力系统的可达性和常返性的多项式时间算法（基于${\bf Z}_{p}$上的线性代数）.有趣的是，我们提供了一个反例，表明这种连接并不扩展到所有的图。
英文摘要：
We study a dynamical system motivated by our earlier work on the statistical physics of social balance on graphs that can be viewed as a generalization of annihilating walks along two directions: first, the interaction topology is a hypergraph; second, the ``number of particles`` at a vertex of the hypergraph is an element of a finite field ${\bf Z}_{p}$ of integers modulo $p$, $p\geq 3$. Equivalently, particles move on a hypergraph, with a moving particle at a vertex being replaced by one indistinguishable copy at each neighbor in a given hyperedge; particles at a vertex collectively annihilate when their number reaches $p$. The system we study can also be regarded as a natural generalization of certain lights-out games to finite fields and hypergraph topologies. Our result shows that under a liberal sufficient condition on the nature of the interaction hypergraph there exists a polynomial time algorithm (based on linear algebra over ${\bf Z}_{p}$) for deciding reachability and recurrence of this dynamical system. Interestingly, we provide a counterexample that shows that this connection does not extend to all graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
