Mathematics – Combinatorics
Scientific paper
2012-03-07
Mathematics
Combinatorics
31 pages, 3 figures
Scientific paper
The aim of this paper is to study the behaviour of rotor-router walks on directed covers of finite graphs. The latter are also called in the literature trees with finitely many cone types or periodic trees. A rotor-router walk is a deterministic version of a random walk, in which the walker is routed to each of the neighbouring vertices in some fixed cyclic order. We study several quantities related to rotor-router walks such as: order of the rotor-router group, order of the root element in the rotor-router group and the connection with random walks. For random initial configurations of rotors, we also address the question of recurrence and transience of transfinite rotor-router walks. On homogeneous trees, the recurrence/transience was studied by Angel and Holroyd. We extend their theory and provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree.
Huss Wilfried
Sava Ecaterina
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