On Playing Golf with Two Balls
SIAM Journal on Discrete Mathematics
Simulating a Random Walk with Constant Error
Combinatorics, Probability and Computing
Deterministic random walks on the integers
European Journal of Combinatorics
Deterministic random walks on regular trees
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The rotor--router model on regular trees
Journal of Combinatorial Theory Series A
European Journal of Combinatorics
Deterministic random walks on the two-dimensional grid
Combinatorics, Probability and Computing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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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 neighboring 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 toward the root. For independently and identically distributed 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.