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

2. 12341234

3. 12341234 43214321

4. 12341234 21342134 43214321

5. 12341234 21342134 23142314 43214321

6. 12341234 21342134 23142314 23412341 43214321

7. 12341234 21342134 23142314 23412341 32413241 43214321

8. 12341234 21342134 23142314 23412341 32413241 34213421 43214321

9. 12341234 21342134 23142314 23412341 32413241 34213421 43214321

10. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 To get from 1 n to n 1 requires N:= nearest-neighbour swaps

11. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 any route from 1 n to n 1 in exactly N:= nearest-neighbour swaps A Sorting Network = E.g. n=4:

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. 12341234 21342134 23142314 23412341 32413241 34213421 43214321

14. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations

15. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations particle trajectory

16. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations particle trajectory 23412341 permutation matrix (at half-time) 9 efficient simulation algorithm for USN... 1 2 3 4

17. Swap locations, n=100

18. Swap locations, n=2000

19. 12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations s1=1s1=1 s2=2s2=2 s3=3s3=3 s4=1s4=1 s5=2s5=2 s6=1s6=1

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

21. In progress: Process of first k swaps in positions cn...cn+k as n !1 not depending on c 2 (0,1) ! random limit

22. Proof of stationarity: 12341234 21342134 23142314 23412341 32413241 34213421 43214321

23. 21342134 23142314 23412341 32413241 34213421 43214321

24. 12341234 13241324 13421342 31423142 34123412 43124312

25. 12341234 13241324 13421342 31423142 34123412 43124312

26. 12341234 13241324 13421342 31423142 34123412 43124312 43214321

27. 12341234 13241324 13421342 31423142 34123412 43124312 43214321 (s 1,...,s N ) (s 2,...,s N,n-s 1 ) is a bijection from {sorting networks} to itself. So for USN:

28. Selected trajectories, n=2000

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

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

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

32. Conjecture: scaled permutation matrix at time N/2 Archimedes measure projection of surface measure on S 2 ½ R 3 to R 2 (unique circularly symmetric measure with uniform linear projections; on x 2 +y 2 <1 ) scaled permutation matrix at time tN Arch. meas.

33. Theorem: scaled permutation matrix at time tN is supported within a certain octagon w.h.p. as n !1 (1-½ p 3- )n

34. Some Proofs: Key tools: 1.Bijection (Edelman-Greene 1987) {sorting networks} $ {standard staircase Young tableaux} 2.New result for profile of random staircase Young tableau (deduced from Pittel-Romik 2006)

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

36. Standard staircase Young tableau: 12 56 84 3 9 710 Fill with 1,,N so each row/col increasing

37. Edelman-Greene algorithm: 12 56 84 3 9 710 1. Remove largest entry

38. Edelman-Greene algorithm: 12 56 84 3 9 7 1. Remove largest entry

39. Edelman-Greene algorithm: 12 56 84 3 9 7 2. Replace with larger of neighbours " Ã

40. Edelman-Greene algorithm: 12 56 84 3 9 7 2. Replace with larger of neighbours " Ã

41. Edelman-Greene algorithm: 12 56 84 3 9 7 2. Replace with larger of neighbours " Ã...repeat

42. Edelman-Greene algorithm: 1 2 56 84 3 9 7 2. Replace with larger of neighbours " Ã...repeat

43. Edelman-Greene algorithm: 1 2 56 84 3 9 7 3. Add 0 in top corner 0

44. Edelman-Greene algorithm: 2 3 67 95 4 10 8 4. Increment 1

45. Edelman-Greene algorithm: 2 3 67 95 48 5. Repeat everything... 1

46. Edelman-Greene algorithm: 2 3 67 95 4 8 5. Repeat everything... 1

47. Edelman-Greene algorithm: 2 3 67 95 4 8 5. Repeat everything... 1

48. Edelman-Greene algorithm: 2 3 67 95 4 8 5. Repeat everything... 1

49. Edelman-Greene algorithm: 2 3 67 95 4 8 5. Repeat everything... 1 0

50. Edelman-Greene algorithm: 3 4 78 106 5 9 5. Repeat everything... 2 1

51. Edelman-Greene algorithm: 3 4 78 6 5 9 5. Repeat everything... 2 1

52. Edelman-Greene algorithm: 4 5 89 7 6 10 5. Repeat everything... 3 21

53. Edelman-Greene algorithm: 4 5 89 7 6 5. Repeat everything... 3 21

54. Edelman-Greene algorithm:

55. etc

56. Edelman-Greene Theorem: After N steps, 45 o

57. Edelman-Greene Theorem: After N steps, get swap process of a sorting network! 1 1 1 1 4 4 4 4 5 5 5 2 2 4 4 1 1 5 5 4 4 4 3 4 2 2 2 5 1 2 2 2 3 4 3 3 5 5 2 2 1 1 3 2 5 5 5 3 3 3 3 3 3 1 1

58. Edelman-Greene Theorem: After N steps, get swap process of a sorting network And this is a bijection! And can explicitly describe inverse!

59. Theorem (Pittel-Romik): For a uniform random n x n square tableau, 9 limiting shape with contours: Corollary (AHRV): For uniform random staircase tableau, limiting shape is half of this. (Proof uses Greene-Nijenhuis-Wilf Hook Walk)

60. Proof of LLN (swap process ) semic. x Leb.) Swaps in space-time window [an,bn]x[0, N] come from entries >(1- )N in tableau: an bn entries exiting in [an,bn] µ µ # ¼ area under contour ¼ semicircle

61. Proof of octagon and Holder bounds Inverse Edelman-Greene bijection ( ¼ RSK algorithm) ) # entries <k in 1 st row ¸ longest % subseq. of swaps by time k ¸ furthest any particle moves up by time k So can bound this using limit shape.

62. Why do we believe the conjectures? The permutahedron: embedding of Cayley graph (S n, n.n. swaps) in R n : -1 =( -1 (1),..., -1 (n)) 2 R n n=4: embeds in an (n-2)-sphere n=5

63. Theorem: If a non-random sorting network lies close to some great circle on the permutahedron, then: 1.Trajectories ¼ Sine curves 2. Half-time permutation ¼ Archimedes measure 3. Swap process ¼ semicircle x Lebesgue

64. Theorem: If a non-random sorting network lies close to some great circle on the permutahedron, then: 1.Trajectories ¼ Sine curves 2. Half-time permutation ¼ Archimedes measure 3. Swap process ¼ semicircle x Lebesgue o(n) in | | 1 Simulations suggest O( p n) for USN!

65. Proof of Theorem: close to great circle ) ¼ Sine trajectories (up to a time change), ¼ rotating disc calculation ) semicircle law swap rate uniform ) rotation uniform ) no time change projections uniform ) ¼ Archimedes

66. Stretchable sorting network: 1 2 3 4 (, rotating disc) USN not stretchable w.h.p. as n !1 Main conj. ) - USN ¼ stretchable w.h.p. - sub-network of k random items is stretchable w.h.p. 1 2 3 4

67. N.B. Not every sorting network lies close to a great circle! E.g. typical network through n/2... 1 n... n/2+1 1... n/2 n/2+1... n... n/2+1 n/2... 1 USN (But this permutation is very unlikely).

68. Uniform swap model...