A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two noncolliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) any d-regular graph avoiding a fixed subgraph depending on d; and (ii) any square-free graph with minimum degree at least three. A corollary of the first result is that a uniformly random regular graph on n vertices admits an avoidance coupling with high probability.

中文翻译：

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非完全图上的避免耦​​合

同一有限图上的随机游走者的耦合，他们依次轮流，如果游走者从不发生碰撞，则称其为回避耦合。以前对这些过程的研究几乎完全集中在完整图上，特别是避免耦合可以包含多少步行者。对于其他图，除了特殊情况，两个非碰撞的简单随机游走器是否可以耦合一直是个悬而未决的问题。在本文中，我们在 (i) 任何d -正则图上构建这样的耦合，避免依赖于d的固定子图；(ii) 最小度数至少为 3 的任何无平方图。第一个结果的推论是n上的均匀随机正则图 顶点以高概率承认避免耦合。