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RSK asymptotic determinism of this and that the key result: new box proof of the key result

Asymptotic determinism of

Robinson-Schensted-Knuth algorithm

joint work with Dan Romik

Piotr �niady


Robinson-Schensted-Knuth algorithm � induction step

5 37 41 82

23 53 69

74 99

insertion tableau P(x)

x = (23, 53, 74, 5, 99, 69, 82, 37, 41)



5 37 41 82

23 53 69

74 99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 37 41 82

23 53 69

74 99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 37 41 82

23 53 69

74 99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 37 41 82

23 53 69

74 99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 53 69

74 99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 53 69

74 99

18


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 53 69

74 99

18


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 53 69

74 99

18


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 53 69

74 99

18


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

74 99

18


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

74 99

18

37


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

74 99

18

37


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

74 99

18

37


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

74 99

18

37


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37

53


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37

53


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37

53


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37

53

74


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 41 82

23 69

99

18

37

53

74bumping route


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 18 41 82

23 37 69

53 99

74


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)




x = ∅



23


x = (23)



23 53


x = (23, 53)



23 53 74


x = (23, 53, 74)



53 745

23


x = (23, 53, 74, 5)



5 53 74

23

99


x = (23, 53, 74, 5, 99)



5 53 99

23

69

74


x = (23, 53, 74, 5, 99, 69)



5 53 69

23 74

82

99


x = (23, 53, 74, 5, 99, 69, 82)



5 69 82

23 99

37

53

74


x = (23, 53, 74, 5, 99, 69, 82, 37)



5 37 82

23 53

74

41

69

99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41)



5 41 82

23 69

99

18

37

53

74


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18)



5 18 82

23 37

53

74

39

41

69

99


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18, 39)



5 18 39

23 37 41

53 69

74 99

61

82


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18, 39, 61)



5 18 39 61

23 37 41 82

53 69

74 99

73


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18, 39, 61, 73)



5 18 39 61

23 37 41

53 69

74 99

66

73

82


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18, 39, 61, 73, 66)



5 18 61 66

23 37 73

69 82

99

22

39

41

53

74


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18, 39, 61, 73, 66, 22)


outlook

x1, x2, . . . independent random variableswith uniform distribution on the interval [0, 1];

insertion tableau Pm = P(x1, . . . , xm);

General problem

What can we say about (the time evolution of)

the insertion tableau Pm?

�with the right scaling of time and space,

the answer is deterministic (asymptotically)�


di�usion of a box

xn (Pm) denotes the location of the box containing xnin the insertion tableau Pm, for m ≥ n;

Concrete problem 1

Suppose that n and xn are known;

what can we say about the time evolution of xn (Pm)

for m = n, n + 1, . . . ?


di�usion of a box


di�usion of a box


di�usion of a box

xn (Pm) denotes the location of the box containing xnin insertion tableau Pm, for m ≥ n;

Theorem

There exists an explicit function G : R+ → R2+ such that

xn (Pbneτ c)√n xn

in probability−−−−−−−→n→∞

Gτ for τ ≥ 0.


hydrodynamic limit of RSK


bumping routes

5 41 82

23 69

99

18

37

53

74bumping route


x = (23, 53, 74, 5, 99, 69, 82, 37, 41, 18︸︷︷︸xn

)


bumping routes

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Theorem

Bumping route (scaled by factor 1√n xn

)

obtained by adding entry xn to the tableau Pn−1converges in probability (as n→∞) to a deterministic curve Gτ .


the key result: new box

P(x1, . . . , xn, xn+1) \ P(x1, . . . , xn) ={ }

Theorem∥∥∥∥∥√n − (RSKcos xn+1,RSKsin xn+1)

∥∥∥∥∥ n→∞−−−−−−−→in probability

0


new box

2

2xn+1 = 0.0

xn+1 = 0.2

xn+1 = 0.4

xn+1 = 0.6

xn+1 = 0.8

xn+1 = 1.0


new box

2

2xn+1 = 0.0

xn+1 = 0.2

xn+1 = 0.4

xn+1 = 0.6

xn+1 = 0.8

xn+1 = 1.0


the key result explains the behavior of bumping routes

2

2

Young diagram with n boxes, scaled by 1√n



2

2


(RSKcos xn,RSKsin xn)



2

2





2

2


r = const

(r RSKcosφ, r RSKsinφ)




2

2


r = const

(r RSKcosφ, r RSKsinφ)

exercise: φ = xnr2



proof, part 1 � reduction of problem

instead of (for deterministic xn+1)

P(x1, . . . , xn, xn+1 ) \ P(x1, . . . , xn) ={ }



we study (for random 0 < t1 < · · · < tk < 1)

P(x1, . . . , xn, t1, . . . , tk ) \ P(x1, . . . , xn) ={

1 ,. . . , k}

1

2

. . . k



we study (for random 0 < t1 < · · · < tk < 1)

P(x1, . . . , xn, t1, . . . , tk ) \ P(x1, . . . , xn) ={

1 ,. . . , k}

1

2

. . . k

if xn+1 < ti then is north-west from i

for ik≈ xn+1 + ε, this happens with high probability, as k →∞


representations of the symmetric groups

representation ρ of a group G is a homomorphism to matrices

ρ : G → GLk

irreducible representation ρλ

of the symmetric group Sn

←→ Young diagram λwith n boxes

Littlewood-Richardson coe�cients(ρλ ⊗ ρµ

)xS|λ|+|µ|

S|λ|×S|µ|

=⊕ν

cνλ,µ ρν


RSK and Littlewood-Richardson coe�cients

if 0 ≤ x1, . . . , xn ≤ 1 is a random sequence, such that

shape of P(x1, . . . , xn) = λ;

and 0 ≤ t1, . . . , tk ≤ 1 is a random sequence, such that

shape of P(t1, . . . , tk) = µ

then the random Young diagram

shape of P(x1, . . . , xn, t1, . . . , tk)

has the same distribution as random irreducible component of

V λ ⊗ V µxSn+kSn×Sk




shape of P(x1, . . . , xn) = λ;

and 0 ≤ t1, . . . , tk ≤ 1 is a random sequence, such that

shape of P(t1, . . . , tk) = (k) =


shape of P(x1, . . . , xn, t1, . . . , tk)


V λ ⊗ V (k)xSn+kSn×Sk




shape of P(x1, . . . , xn) = λ;

and 0 ≤ t1 < · · · < tk ≤ 1 is a random sequence, such that

shape of P(t1, . . . , tk) = (k) =


shape of P(x1, . . . , xn, t1, . . . , tk)


V λ ⊗ V (k)xSn+kSn×Sk


content of the box

content(�) = (x-coordinate)− (y -coordinate)

Example

0 1 2 3 4

−1 0 1 2 3

−2

c

content of Young diagram = (−2,−1, 0, 0, 1, 1, 2, 3)


Jucys�Murphy elements

Xi = (1, i) + (2, i) + · · ·+ (i − 1, i) for i ∈ {1, . . . , n}

X1, . . . ,Xn are elements of the symmetric group algebra C(Sn)

for any Young diagram λ with contents (c1, . . . , cn)and a symmetric polynomial P(x1, . . . , xn)

χλ(P(X1, . . . ,Xn)

)=

Tr ρλ(P(X1, . . . ,Xn)

)Tr ρλ(1)

= ?


Jucys�Murphy elements

Xi = (1, i) + (2, i) + · · ·+ (i − 1, i) for i ∈ {1, . . . , n}

X1, . . . ,Xn are elements of the symmetric group algebra C(Sn)

for any Young diagram λ with contents (c1, . . . , cn)and a symmetric polynomial P(x1, . . . , xn)

χλ(P(X1, . . . ,Xn)

)=

Tr ρλ(P(X1, . . . ,Xn)

)Tr ρλ(1)

= P(c1, . . . , cn)


growth of Young diagrams and Jucys-Murphy elements

c

let λ ` n, µ ` k be �xed Young diagrams

let Γ be a random irreducible component ofV λ ⊗ V µ

xSn+kSn×Sk

let cn+1, . . . , cn+k be the contents of boxesof Γ \ λ

then for any symmetric polynomialP(xn+1, . . . , xn+k) we have(

χλ ⊗ χµ)(

P(Xn+1, . . . ,Xn+k)ySn+kSn×Sk

)= EP(cn+1, . . . , cn+k)


proof, part 2

if k ≈ 4√n

1k

(δ c1√

n

+ · · ·+ δ ck√n

)in probability−−−−−−−→

n→∞µSC =

2-2

,

where ci = c( i )

Hint: p-th moment of the left-hand-side

1k

∑j

(cj√n

)p

is a random variable,show that the mean converges to p-th moment of µSCshow that the variance converges to zero


proof, part 2

if k ≈ 4√n

1k

(δ c1√

n

+ · · ·+ δ ck√n

)in probability−−−−−−−→

n→∞µSC =

2-2

,

where ci = c( i )

since c1 < · · · < ck , this implies that if ik→ xn+1 then

c( i )√n

in probability−−−−−−−→ F−1µSC(xn+1)


proof, part 3

shape of Pn (scaled by factor 1√n)

with high probability concentratesaround some explicit shape

Logan, Shepp, Vershik, Kerov

√nis with high probability close

to the boundary of this limit shape


further reading

Dan Romik, Piotr �niadyJeu de taquin dynamics on in�nite Young tableaux andsecond class particlesAnnals of Probability 43 (2015), no. 2, 682�737

Dan Romik, Piotr �niadyLimit shapes of bumping routes in the Robinson-SchenstedcorrespondenceRandom Structures Algorithms 48 (2016), no. 1, 171�18

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This presentation contains animations which require PDF ...psniady.impan.pl/data/uploads/slajdy/... · accepts JavaScript. For best results use Acrobat Reader. RSK asymptotic determinism