1. Rotor routing on 2-manifolds
Taro Shima Nicholas Tomlin
Brown University
SUMS, March 2016
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2. Outline
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3. Outline
Definition of rotor routing
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4. Outline
Definition of rotor routing
Rotor routing on Z2
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5. Outline
Definition of rotor routing
Rotor routing on Z2
Recurrence and transience
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6. Outline
Definition of rotor routing
Rotor routing on Z2
Recurrence and transience
Rotor routing on other 2-manifolds
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7. Outline
Definition of rotor routing
Rotor routing on Z2
Recurrence and transience
Rotor routing on other 2-manifolds
Visualizations and experimental results
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8. What is rotor routing?
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9. What is rotor routing?
Rotor routing is a deterministic version of the random walk.
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10. What is rotor routing?
Definition (Rotor)
An arrow pointing from one vertex to another.
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11. What is rotor routing?
Definition (Rotor)
An arrow pointing from one vertex to another.
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Definition (Rotor Configuration)
An assignment of a single rotor to every vertex on a graph.
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12. What is rotor routing?
Definition (Rotor Configuration)
An assignment of a single rotor to every vertex on a graph.
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13. What is rotor routing?
Definition (Rotor Configuration)
An assignment of a single rotor to every vertex on a graph.
Definition (Rotor Mechanism)
Given a set of possible rotors E starting at a given point, a rotor
mechanism is a cyclic permutation of E.
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14. What is rotor routing?
Definition (Rotor Configuration)
An assignment of a single rotor to every vertex on a graph.
Definition (Rotor Mechanism)
Given a set of possible rotors E starting at a given point, a rotor
mechanism is a cyclic permutation of E.
Definition (Rotor Walk)
A traversal of a particle along a initial rotor configuration. The particle
follows the direction of rotors, which then change direction according to
the rotor mechanism.
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15. Rotor walk example
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16. Rotor walk example
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17. Rotor walk example
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18. Rotor walk example
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19. Rotor walk example
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20. Rotor walk example
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21. Rotor walk example
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22. Rotor routing on Z2
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23. Rotor routing on Z2
Definition (Rotor mechanism on Z2
)
There are four possible rotors starting at a given point:
E = {North, East, South, West}.
Rotors turn counter-clockwise, repeatedly cycling through E.
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24. Rotor routing on Z2
Definition (Rotor mechanism on Z2
)
There are four possible rotors starting at a given point:
E = {North, East, South, West}.
Rotors turn counter-clockwise, repeatedly cycling through E.
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25. Recurrence and transience
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26. Recurrence and transience
Definition (Recurrence)
A walk is recurrent if it visits each point infinitely many times.
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27. Recurrence and transience
Definition (Recurrence)
A walk is recurrent if it visits each point infinitely many times.
Random walk on Z, Z2
Repeated rotor walks on Z, Z2 with single-direction configuration
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28. Recurrence and transience
Definition (Recurrence)
A walk is recurrent if it visits each point infinitely many times.
Random walk on Z, Z2
Repeated rotor walks on Z, Z2 with single-direction configuration
Definition (Transience)
A walk is transient if it visits each point finitely many times.
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29. Recurrence and transience
Definition (Recurrence)
A walk is recurrent if it visits each point infinitely many times.
Random walk on Z, Z2
Repeated rotor walks on Z, Z2 with single-direction configuration
Definition (Transience)
A walk is transient if it visits each point finitely many times.
Random walk on Zd where d 3
Repeated rotor walks on Z3 with single-direction configuration
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30. Theorems about recurrence and transience
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31. Theorems about recurrence and transience
Theorem (Angel-Holroyd 2011)
Given a rotor mechanism and a configuration, a single rotor walk is either
recurrent for every starting vertex, or transient for every starting vertex.
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32. Theorems about recurrence and transience
Theorem (Angel-Holroyd 2011)
Given a rotor mechanism and a configuration, a single rotor walk is either
recurrent for every starting vertex, or transient for every starting vertex.
Corollary
Given a rotor mechanism, two rotor configurations di↵ering at only finitely
many vertices are either both recurrent or both transient.
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33. Theorems about recurrence and transience
Corollary
Given a rotor mechanism, two rotor configurations di↵ering at only finitely
many vertices are either both recurrent or both transient.
Proof
We need only consider two rotor configurations r, r0 di↵ering at only one
vertex v, such that a single rotation of the rotor at v produces r0 from r.
If r is recurrent, a rotor walk started at v produces r0 with the particle
moved to some other vertex. By the aforementioned theorem, r0 must also
be recurrent.
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34. Random configuration on Z2
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35. Random configuration on Z2
Conjecture (Levine-Peres 2015)
Given a random configuration of rotors on Z2, a rotor walk on that
configuration is recurrent with probability 1.
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36. Random configuration on Z2
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37. Rotor routing on other 2-manifolds
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38. Rotor routing on other 2-manifolds
Definition (2-manifold)
A topological space whose points all have open disks as neighborhoods.
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39. Rotor routing on other 2-manifolds
Definition (2-manifold)
A topological space whose points all have open disks as neighborhoods.
Some 2-manifolds:
Cylinder
M¨obius strip
Torus
Sphere
Klein bottle
Real projective plane
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40. Rotor routing on other 2-manifolds
Definition (Compactness)
If a space is compact, every open cover has a finite sub-cover.
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41. Rotor routing on other 2-manifolds
Definition (Compactness)
If a space is compact, every open cover has a finite sub-cover.
Non-compact 2-manifolds:
Plane
Cylinder
M¨obius strip
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42. Rotor routing on other 2-manifolds
Definition (Compactness)
If a space is compact, every open cover has a finite sub-cover.
Non-compact 2-manifolds:
Plane
Cylinder
M¨obius strip
Compact 2-manifolds:
Torus
Sphere
Klein bottle
Real projective plane
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43. Rotor routing on other 2-manifolds
A gluing diagram shows how edges of a polygon are connected to create a
surface. When a particle hits an edge that is associated to another, the
particle ”reappears” on the associated edge.
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44. Experimental results
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45. Experimental results
Conjecture
Repeated rotor walks on the M¨obius strip, cylinder, and plane will
eventually create congruent octagonal patterns for the single-direction
configuration.
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46. Experimental results
Conjecture
Repeated rotor walks on the M¨obius strip, cylinder, and plane will
eventually create congruent octagonal patterns for the single-direction
configuration.
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47. Experimental results
Observation
For any rotor configuration and rotor mechanism, a rotor walk on a
compact surface eventually results in a finite loop of configurations.
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48. Future work
Testing alternate configurations, including di↵erent lattices and
transient configurations.
Experimenting with rotor routing in higher dimensions.
Developing a predictive model for surface type based on loop length
on compact surfaces.
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