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Rotor walks on general trees
The rotor walk on a graph is a deterministic analogue of random walk. Each
vertex is equipped with a rotor, which routes the walker to the neighbouring
vertices in a fixed cyclic order on successive visits. We consider rotor walk
on an infinite rooted tree, restarted from the root after each escape to
infinity. We prove that the limiting proportion of escapes to infinity equals
the escape probability for random walk, provided only finitely many rotors send
the walker initially towards the root. For i.i.d. random initial rotor
directions on a regular tree, the limiting proportion of escapes is either zero
or the random walk escape probability, and undergoes a discontinuous phase
transition between the two as the distribution is varied. In the critical case
there are no escapes, but the walker's maximum distance from the root grows
doubly exponentially with the number of visits to the root. We also prove that
there exist trees of bounded degree for which the proportion of escapes
eventually exceeds the escape probability by arbitrarily large o(1) functions.
No larger discrepancy is possible, while for regular trees the discrepancy is
at most logarithmic.Comment: 32 page
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