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Recurrence and range of rotor walk

Let $d>1$ . For every node $x$ in the lattice $\mathbb Z^d$ , fix a cyclic ordering of the neighbors of $x$ . E.g., for $d=2$ , we can use the ordering NESW. The rotor walk is obtained by fixing an initial configuration of arrows pointing from each vertex of the lattice to one of its neighbors. We start a particle at the origin, and each time it reaches a vertex $x$ , we first move the arrow from the current position to the next one according to the prescribed ordering, and then move the particle according to the arrow.

For a uniformly random initial configuration of the arrows, will the particle be recurrent (i.e., will it return to its starting point infinitely often almost surely) or not?
It is natural to conjecture recurrence for $d=2$ and transience for $d>2$ .

Another quantity of interest is the Range $R_t$ of the walk, i.e., the number of nodes visited by time $t$ . In [1] a Heuristic argument was presented that $R_t$ should be typically of order $t^{2/3}$ in two dimensions. A lower bound of this order was proved in [2] for all initial configurations, but no upper bound of the form $o(t)$ (for the random initial configuration) is known. Indeed, such a bound would imply recurrence. In dimension $d \ge 3$ , it is expected that $R_t$ is of order $t$ .

For results on other lattices, see [2] and [3].

REFERENCES

[1] V. B. Priezzhev, D. Dhar, A. Dhar, S. Krishnamurthy, Eulerian walkers as a model of self-organized criticality, Phys. Rev. Lett. 77 (1996) 5079--5082

[2] Florescu, Laura, Lionel Levine, and Yuval Peres. "The range of a rotor walk." The American Mathematical Monthly 123, no. 7 (2016): 627-642.

[3] Chan, Swee Hong. "Recurrence of horizontal-vertical walks." arXiv preprint arXiv:2012.10811 (2020). To appear, Ann. Institut H. Poincar\'e, Probab. Stat.

Recurrence and range of rotor walk

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