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Omer Angel

Random Sorting Networks. Omer Angel. Alexander Holroyd. Dan Romik. Balint Virag. math.PR/0609538. Adv. in Math. 2007. Thanks to: Nathanael Berestycki, Alex Gamburd, Alan Hammond, Pawel Hitczenko, Martin Kassabov, Rick Kenyon, Scott Sheffield, David Wilson, Doron Zeilberger. 1 2 3

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Omer Angel

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  1. Random Sorting Networks Omer Angel Alexander Holroyd Dan Romik BalintVirag math.PR/0609538 Adv. in Math. 2007 Thanks to: Nathanael Berestycki, Alex Gamburd, Alan Hammond, Pawel Hitczenko, Martin Kassabov, Rick Kenyon, Scott Sheffield, David Wilson, Doron Zeilberger

  2. 1 2 3 4

  3. 4 3 2 1 1 2 3 4

  4. 2 1 3 4 4 3 2 1 1 2 3 4

  5. 2 1 3 4 2 3 1 4 4 3 2 1 1 2 3 4

  6. 2 1 3 4 2 3 1 4 2 3 4 1 4 3 2 1 1 2 3 4

  7. 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 4 3 2 1 1 2 3 4

  8. 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4

  9. 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4

  10. 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4 To get from 1Ln to nL1 requires N:= nearest-neighbour swaps

  11. E.g. n=4: 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4 A Sorting Network = any route from 1Ln to nL1 in exactly N:= nearest-neighbour swaps

  12. Theorem (Stanley 1984). # of n-particle sorting networks = Uniform Sorting Network (USN): choose an n-particle sorting network uniformly at random. E.g. n=3:

  13. 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4

  14. swap locations 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4

  15. swap locations 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4 particle trajectory

  16. 2 3 4 1 swap locations 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4 particle trajectory permutation matrix (at half-time) 1 2 3 4 9 efficient simulation algorithm for USN...

  17. Swap locations, n=100

  18. Swap locations, n=2000

  19. swap locations 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4 s1 = 1 s2 = 2 s3 = 3 s4 = 1 s5 = 2 s6 = 1

  20. Theorem:For USN: 1. Sequence of swap locations (s1,...,sN) is stationary 2. Scaled first swap location 3. Scaled swap process 8 n as n!1 as n!1 (Note: not true for all sorting networks, e.g. bubble sort)

  21. In progress: Process of first k swaps in positions cn...cn+k ! random limit as n!1 not depending on c2(0,1)

  22. Proof of stationarity: 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1 1 2 3 4

  23. Proof of stationarity: 2 1 3 4 2 3 1 4 2 3 4 1 3 2 4 1 3 4 2 1 4 3 2 1

  24. Proof of stationarity: 1 2 3 4 1 3 2 4 1 3 4 2 3 1 4 2 3 4 1 2 4 3 1 2

  25. Proof of stationarity: 1 2 3 4 1 3 2 4 1 3 4 2 3 1 4 2 3 4 1 2 4 3 1 2

  26. Proof of stationarity: 1 2 3 4 1 3 2 4 1 3 4 2 3 1 4 2 3 4 1 2 4 3 1 2 4 3 2 1

  27. So for USN: Proof of stationarity: 1 2 3 4 1 3 2 4 1 3 4 2 3 1 4 2 3 4 1 2 4 3 1 2 4 3 2 1 (s1,...,sN) a (s2,...,sN,n-s1) is a bijection from {sorting networks} to itself.

  28. Selected trajectories, n=2000

  29. Scaled trajectory of particle i: Ti:[0,1]![-1,1] 1 i -1 1 0

  30. Conjecture: trajectories ! random Sine curves: as n!1 (random) Theorem: scaled trajectories have subsequential limits which are a.s. Hölder(½). as n!1

  31. Half-time permutation matrix, n=2000 animation

  32. projection of surface measure on S2½R3 to R2 (unique circularly symmetric measure with uniform linear projections; on x2+y2<1 ) scaled permutation matrix at time tN Arch. meas. Conjecture: scaled permutation matrix at time N/2 Archimedes measure

  33. (1-½p3-e)n Theorem: scaled permutation matrix at time tN is supported within a certain octagon w.h.p. as n!1

  34. Some Proofs: • Key tools: • Bijection (Edelman-Greene 1987) • {sorting networks} $ {standard staircase • Young tableaux} • New result for • profile of random staircase Young tableau • (deduced from Pittel-Romik 2006)

  35. Staircase Young diagram: (E.g. n=5) n-1 N cells

  36. Standard staircase Young tableau: 1 2 4 8 3 5 6 7 10 9 Fill with 1,L,N so each row/col increasing

  37. Edelman-Greene algorithm: 1 2 4 8 3 5 6 7 10 9 1. Remove largest entry

  38. Edelman-Greene algorithm: 1 2 4 8 3 5 6 7 9 1. Remove largest entry

  39. Edelman-Greene algorithm: 1 2 4 8 3 5 6 7 9 2. Replace with larger of neighbours "Ã

  40. Edelman-Greene algorithm: 1 2 4 8 3 5 6 7 9 2. Replace with larger of neighbours "Ã

  41. Edelman-Greene algorithm: 1 2 4 8 5 6 3 7 9 2. Replace with larger of neighbours "Ã ...repeat

  42. Edelman-Greene algorithm: 2 4 8 1 5 6 3 7 9 2. Replace with larger of neighbours "Ã ...repeat

  43. Edelman-Greene algorithm: 0 2 4 8 1 5 6 3 7 9 3. Add 0 in top corner

  44. Edelman-Greene algorithm: 1 3 5 9 2 6 7 4 8 10 4. Increment

  45. Edelman-Greene algorithm: 1 3 5 9 2 6 7 4 8 5. Repeat everything...

  46. Edelman-Greene algorithm: 1 3 5 9 2 6 7 8 4 5. Repeat everything...

  47. Edelman-Greene algorithm: 1 3 5 9 6 7 2 8 4 5. Repeat everything...

  48. Edelman-Greene algorithm: 3 5 9 6 7 1 2 8 4 5. Repeat everything...

  49. Edelman-Greene algorithm: 0 3 5 9 6 7 1 2 8 4 5. Repeat everything...

  50. Edelman-Greene algorithm: 1 4 6 10 7 8 2 3 9 5 5. Repeat everything...

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