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Omer AngelAlexander Holroyd Dan RomikBalint Virag math.PR/ 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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To get from 1 n to n 1 requires N:= nearest-neighbour swaps

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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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swap locations

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swap locations particle trajectory

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swap locations particle trajectory permutation matrix (at half-time) 9 efficient simulation algorithm for USN

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Swap locations, n=100

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Swap locations, n=2000

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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:

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(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: Fill with 1,,N so each row/col increasing

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Edelman-Greene algorithm: Remove largest entry

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Edelman-Greene algorithm: Remove largest entry

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Edelman-Greene algorithm: Replace with larger of neighbours " Ã

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Edelman-Greene algorithm: Replace with larger of neighbours " Ã

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Edelman-Greene algorithm: Replace with larger of neighbours " Ã...repeat

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Edelman-Greene algorithm: Replace with larger of neighbours " Ã...repeat

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Edelman-Greene algorithm: Add 0 in top corner 0

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Edelman-Greene algorithm: Increment 1

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Edelman-Greene algorithm: Repeat everything... 1

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Edelman-Greene algorithm: Repeat everything... 1

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Edelman-Greene algorithm: Repeat everything... 1

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Edelman-Greene algorithm: Repeat everything... 1

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Edelman-Greene algorithm: Repeat everything

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Edelman-Greene algorithm: Repeat everything

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Edelman-Greene algorithm: Repeat everything

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Edelman-Greene algorithm: Repeat everything

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Edelman-Greene algorithm: Repeat everything

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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!

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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

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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: (, 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

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

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Uniform swap model...

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