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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

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12341234

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12341234 43214321

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12341234 21342134 43214321

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12341234 21342134 23142314 43214321

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12341234 21342134 23142314 23412341 43214321

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12341234 21342134 23142314 23412341 32413241 43214321

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12341234 21342134 23142314 23412341 32413241 34213421 43214321

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12341234 21342134 23142314 23412341 32413241 34213421 43214321

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

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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:

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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:

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12341234 21342134 23142314 23412341 32413241 34213421 43214321

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12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations

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12341234 21342134 23142314 23412341 32413241 34213421 43214321 swap locations particle trajectory

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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

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Swap locations, n=100

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Swap locations, n=2000

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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

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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)

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

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Proof of stationarity: 12341234 21342134 23142314 23412341 32413241 34213421 43214321

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21342134 23142314 23412341 32413241 34213421 43214321

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12341234 13241324 13421342 31423142 34123412 43124312

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12341234 13241324 13421342 31423142 34123412 43124312

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12341234 13241324 13421342 31423142 34123412 43124312 43214321

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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:

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Selected trajectories, n=2000

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

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

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Half-time permutation matrix, n=2000 animation

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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.

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

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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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Edelman-Greene algorithm:

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etc

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Edelman-Greene Theorem: After N steps, 45 o

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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

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

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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)

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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

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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.

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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

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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

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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!

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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

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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

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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).

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Uniform swap model...

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