Omer AngelAlexander Holroyd Dan RomikBalint Virag math.PR/0609538 Thanks to: Nathanael Berestycki,...

Omer Angel Alexander Holroyd

Dan Romik Balint 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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To get from 1n to n1requires

N:=

nearest-neighbour swaps

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any route from 1n to n1in exactly

N:=

nearest-neighbour swaps

A Sorting Network =

E.g. n=4:

Theorem (Stanley 1984). # of n-particle sorting networks =

Uniform Sorting Network (USN):choose an n-particle sorting networkuniformly at random.

E.g. n=3:

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swaplocations

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swaplocations

particletrajectory

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swaplocations

particletrajectory

2341permutation matrix

(at half-time)

9 efficient simulation algorithm for USN...

1 2 3 4

Swap locations, n=100

Swap locations, n=2000

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swaplocations

s1

=1

s2

=2

s3

=3

s4

=1

s5

=2

s6

=1

Theorem: For USN:

1. Sequence of swap locations (s1,...,sN) is stationary

2. Scaled first swap location

3. Scaled swap process

as n!1

as n!1

8 n

(Note: not true for all sorting networks, e.g. bubble sort)

In progress: Process of first k swapsin positions cn...cn+k

as n!1not depending on c2(0,1)

! random limit

Proof of stationarity:

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(s1,...,sN) (s2,...,sN,n-s1) is a bijectionfrom {sorting networks} to itself.

So for USN:

Selected trajectories, n=2000

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

i

0 1-1

1

Conjecture: trajectories ! random Sine curves:

as n!1

Theorem: scaled trajectories have subsequentiallimits which are a.s. Hölder(½).

as n!1

(random)

Half-time permutation matrix, n=2000

animation

Conjecture: scaled permutation matrix at time N/2

Archimedesmeasure

projection of surfacemeasure on S2 ½ R3 to R2

(unique circularly symmetric measurewith uniform linear projections;

on x2+y2<1 )

scaled permutation matrix at time tN

Arch.meas.

Theorem: scaled permutation matrix at time tNis supported within a certain octagon w.h.p.

as n!1

(1-½p3-)n

Some Proofs:

Key tools:

1. Bijection (Edelman-Greene 1987){sorting networks} $ {standard staircase

Young tableaux}

2. New result forprofile of random staircase Young tableau(deduced from Pittel-Romik 2006)

Staircase Young diagram:

n-1

N cells

(E.g. n=5)

Standard staircase Young tableau:

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5 6

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7 10

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

Edelman-Greene algorithm:

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1. Remove largest entry


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1. Remove largest entry


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2. Replace with larger of neighbours " Ã


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2. Replace with larger of neighbours " Ã


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2. Replace with larger of neighbours " Ã...repeat


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2. Replace with larger of neighbours " Ã...repeat


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3. Add 0 in top corner

0


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4. Increment

1


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4 8

5. Repeat everything...

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etc

Edelman-Greene Theorem:

After N steps,

45o


After N steps,get swap processof a sorting network!

1 1 1 1 4 4 4 4 5 5 52 2 4 4 1 1 5 5 4 4 43 4 2 2 2 5 1 2 2 2 34 3 3 5 5 2 2 1 1 3 25 5 5 3 3 3 3 3 3 1 1


After N steps,get swap processof a sorting network

And this is a bijection!

And can explicitly describe inverse!

Theorem (Pittel-Romik): For a uniform randomn 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)

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

Proof of octagon and Holder bounds

Inverse Edelman-Greene bijection(¼ RSK algorithm) )

# entries <k in 1st row

¸ longest % subseq. of swaps by time k

¸ furthest any particle moves up by time k

So can bound this usinglimit shape.

Why do we believe the conjectures?

The permutahedron: embedding of Cayley graph (Sn, n.n. swaps) in Rn :

-1=(-1(1),...,-1(n))2Rn

n=4:

embedsin an(n-2)-sphere

n=5

Theorem: If a non-random sorting networklies close to some great circle on the permutahedron, then:

1. Trajectories ¼ Sine curves

2. Half-time permutation ¼ Archimedesmeasure

3. Swap process ¼ semicircle x Lebesgue

Theorem: If a non-random sorting networklies close to some great circle on the permutahedron, then:

1. Trajectories ¼ Sine curves

2. Half-time permutation ¼ Archimedesmeasure

3. Swap process ¼ semicircle x Lebesgue

o(n) in | |1Simulations suggestO(pn) for USN!

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

Stretchable sorting network:

1

2

34

(, 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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3

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N.B. Not every sorting network lies close toa great circle! E.g. typical network through

n/2...1n...

n/2+1

1...

n/2n/2+1

...n

n...

n/2+1n/2...1

USN

USN

(But this permutation is very unlikely).

Uniform swap model...

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Omer AngelAlexander Holroyd Dan RomikBalint Virag math.PR/0609538 Thanks to: Nathanael Berestycki,...

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