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Local-to-global principles for the hitting sequence of a rotor walk. (English) Zbl 1243.05105

Summary: In rotor walk on a finite directed graph, the exits from each vertex follow a prescribed periodic sequence. Here we consider the case of rotor walk where a particle starts from a designated source vertex and continues until it hits a designated target set, at which point the walk is restarted from the source.
We show that the sequence of successively hit targets, which is easily seen to be eventually periodic, is in fact periodic. We show moreover that reversing the periodic patterns of all rotor sequences causes the periodic pattern of the hitting sequence to be reversed as well. The proofs involve a new notion of equivalence of rotor configurations, and an extension of rotor walk incorporating time-reversed particles.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C38 Paths and cycles
05C81 Random walks on graphs
05C20 Directed graphs (digraphs), tournaments
90B10 Deterministic network models in operations research
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