论文标题
二维图块组装系统中的确定性非合作结合必须最终具有周期性路径
Deterministic Non-cooperative Binding in Two-Dimensional Tile Assembly Systems Must Have Ultimately Periodic Paths
论文作者
论文摘要
我们在确定性或指示（称此汇合）瓷砖自组装系统中，在二维的确定性或定向（称为Confluent）中考虑了非合件结合，所谓的“温度1”，并显示了该系统具有最终定期组装路径的必要条件。我们证明，当且仅当它包含一个不会与Z2网格中周期性路径相交的无限总成路径时，无限的最大组件最终具有最终周期性的组装路径。此外，我们表明，每个无限组装都必须满足此条件，因此包含最终的周期性路径。该结果是通过超级位置和两个路径的组合获得的，这些路径产生了具有所需属性的新路径，这是我们称之为两种路径的共生的技术。该论文是Arxiv 1901.08575的第一部分的更新和改进的版本。
We consider non-cooperative binding, so-called 'temperature 1', in deterministic or directed (called here confluent) tile self-assembly systems in two dimensions and show a necessary and sufficient condition for such system to have an ultimately periodic assembly path. We prove that an infinite maximal assembly has an ultimately periodic assembly path if and only if it contains an infinite assembly path that does not intersect a periodic path in the Z2 grid. Moreover we show that every infinite assembly must satisfy this condition, and therefore, contains an ultimately periodic path. This result is obtained through a super-position and a combination of two paths that produce a new path with desired properties, a technique that we call co-grow of two paths. The paper is an updated and improved version of the first part of arXiv 1901.08575.
