On the Distribution of Random Staircase Tableaux Pawel Hitczenko - - PowerPoint PPT Presentation

On the Distribution of Random Staircase Tableaux

Pawel Hitczenko & Amanda Parshall June 20, 2014

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Definition of Staircase Tableaux

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Definition of Staircase Tableaux Connections/Motivation

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Definition of Staircase Tableaux Connections/Motivation Previous Results/Conjecture

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Definition of Staircase Tableaux Connections/Motivation Previous Results/Conjecture Results

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Definition of Staircase Tableaux Connections/Motivation Previous Results/Conjecture Results Future Work

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009))

γ α δ β γ δ β γ β α

Figure: A staircase tableau of size 7 and weight α2β3γ3δ2.

Definition A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009))

γ α δ β γ δ β γ β α

Figure: A staircase tableau of size 7 and weight α2β3γ3δ2.

Definition A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:

1 The boxes are empty or contain

an α, β, γ, or δ.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009))

γ α δ β γ δ β γ β α

Figure: A staircase tableau of size 7 and weight α2β3γ3δ2.

Definition A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:

1 The boxes are empty or contain

an α, β, γ, or δ.

2 Every box on the diagonal

contains a symbol.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009))

γ α δ β γ δ β γ β α

Figure: A staircase tableau of size 7 and weight α2β3γ3δ2.

Definition A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:

1 The boxes are empty or contain

an α, β, γ, or δ.

2 Every box on the diagonal

contains a symbol.

3 All boxes in the same column

and above an α or γ are empty.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009))

γ α δ β γ δ β γ β α

Figure: A staircase tableau of size 7 and weight α2β3γ3δ2.

Definition A staircase tableau of size n is a Young diagram of shape (n, n-1, ..., 1) such that:

1 The boxes are empty or contain

an α, β, γ, or δ.

2 Every box on the diagonal

contains a symbol.

3 All boxes in the same column

and above an α or γ are empty.

4 All boxes in the same row and to

the left of an β or δ are empty.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries

Symmetry: Each staircase tableau is symmetric to the staircase tableaux obtained by interchanging the rows and the columns, α’s and β’s, and γ’s and δ’s.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries

Symmetry: Each staircase tableau is symmetric to the staircase tableaux obtained by interchanging the rows and the columns, α’s and β’s, and γ’s and δ’s. Define Sn to be the set of all staircase tableaux of size n.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries

Symmetry: Each staircase tableau is symmetric to the staircase tableaux obtained by interchanging the rows and the columns, α’s and β’s, and γ’s and δ’s. Define Sn to be the set of all staircase tableaux of size n. The weight of S ∈ Sn is the product of all symbols in S: wt(S) = αNαβNβγNγδNδ.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries Cont’d

As proven by Corteel and Dasse-Hartaut: Zn(α, β, γ, δ) :=

S∈Sn wt(S) = n−1

• i=0

(α + β + δ + γ + i(α + γ)(β + δ)).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries Cont’d

As proven by Corteel and Dasse-Hartaut: Zn(α, β, γ, δ) :=

S∈Sn wt(S) = n−1

• i=0

(α + β + δ + γ + i(α + γ)(β + δ)). |Sn| = Zn(1, 1, 1, 1) = 4nn!

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Connections

Introduced due to connections with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Connections

Introduced due to connections with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since then, there have been numerous combinatorical

• connections. E.g. There is a bijection between staircase

tableaux of size n and permutation tableaux of length n + 1 (Corteel & Williams).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Significance of ASEP Connection

A particle model introduced in 1970 (Spitzer).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Significance of ASEP Connection

A particle model introduced in 1970 (Spitzer). Numerous applications including computational biology (Bundschuh), and biochemistry, specifically as a primitive model for protein synthesis (Gibbs, MacDonald, Pipkin).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Significance of ASEP Connection

A particle model introduced in 1970 (Spitzer). Numerous applications including computational biology (Bundschuh), and biochemistry, specifically as a primitive model for protein synthesis (Gibbs, MacDonald, Pipkin). The ASEP “has achieved a paradigmatic status for nonequilibrium systems” (Rajewsky, Santen, Schadschneider, Schreckenberg).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

The ASEP

A Markov Chain with n sites.

• ◦ • • ◦ • ◦ ◦ •

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

The ASEP

A Markov Chain with n sites.

• ◦ • • ◦ • ◦ ◦ •

Transition Probabilities:

• A to • A :

α n + 1

• A to A◦ :

γ n + 1 A ◦ to A• : δ n + 1 A • to A◦ : β n + 1 A • ◦B to A ◦ •B : u n + 1 A ◦ •B to A • ◦B : q n + 1

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Connection with the ASEP

Type of a staircase tableaux:

• for each α or δ on diagonal.
• for each β or γ on diagonal.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Connection with the ASEP

Type of a staircase tableaux:

• for each α or δ on diagonal.
• for each β or γ on diagonal.

Filling rules for u’s and q’s:

1 u’s in all boxes east of a β and

q’s in all boxes east of a δ.

2 u’s in all boxes north of a α or δ

and q’s in all boxes north of a β

• r γ.

γ α δ β γ δ β γ β α

• Pawel Hitczenko & Amanda Parshall

On the Distribution of Random Staircase Tableaux

Connection with the ASEP

Type of a staircase tableaux:

• for each α or δ on diagonal.
• for each β or γ on diagonal.

Filling rules for u’s and q’s:

1 u’s in all boxes east of a β and

q’s in all boxes east of a δ.

2 u’s in all boxes north of a α or δ

and q’s in all boxes north of a β

• r γ.

γ α δ β γ δ β γ β α

• q

u q u q q q u q u q q q q u q q u

Figure: A staircase tableau and it’s type

• • ◦ • ◦ ◦ ◦

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Steady State Probability of the ASEP

Theorem (Corteel and Williams 2010) The steady state probability that the ASEP is in state η is:

• T∈T wt(T)
• S∈S′

n wt(S)

where T is the set of all staircase tableaux of type η, S

′

n is the set

• f all extended staircase tableaux, and wt() is the product of all

symbols in the staircase tableaux.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

α/β-Staircase Tableaux

For combinatorical considerations, let u = q = 1 to obtain the staircase tableaux as defined in the beginning.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

α/β-Staircase Tableaux

For combinatorical considerations, let u = q = 1 to obtain the staircase tableaux as defined in the beginning. W.L.O.G. we can study α/β-staircase tableaux as introduced by Hitczenko and Janson, which are staircase tableaux limited to the symbols α and β.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

α/β-Staircase Tableaux

For combinatorical considerations, let u = q = 1 to obtain the staircase tableaux as defined in the beginning. W.L.O.G. we can study α/β-staircase tableaux as introduced by Hitczenko and Janson, which are staircase tableaux limited to the symbols α and β. Define Sn ⊂ Sn to be the set of all α/β-staircase tableaux.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

α/β-Staircase Tableaux Cont’d

The generating function of α/β-staircase tableaux is: Zn(α, β) :=

• S∈Sn

wt(S) = Zn(α, β, 0, 0) = αnβn(a + b)n where a := α−1 and b := β−1.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

α/β-Staircase Tableaux Cont’d

The generating function of α/β-staircase tableaux is: Zn(α, β) :=

• S∈Sn

wt(S) = Zn(α, β, 0, 0) = αnβn(a + b)n where a := α−1 and b := β−1. Notice that Zn(α, β, γ, δ) = Zn(α + γ, β + δ).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Random Staircase Tableaux

Definition (As by Hitczenko-Janson) For all n ≥ 1, α, β ∈ [0, ∞) with (α, β) = (0, 0), Sn,α,β is defined to be a random α/β-staircase tableau in Sn with respect to the probability distribution on Sn given by: ∀S ∈ Sn, P(Sn,α,β = S) = wt(S) Zn(α, β) = αNαβNβ Zn(α, β).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Random Staircase Tableaux

Definition (As by Hitczenko-Janson) For all n ≥ 1, α, β ∈ [0, ∞) with (α, β) = (0, 0), Sn,α,β is defined to be a random α/β-staircase tableau in Sn with respect to the probability distribution on Sn given by: ∀S ∈ Sn, P(Sn,α,β = S) = wt(S) Zn(α, β) = αNαβNβ Zn(α, β). Extension to Sn: Randomly replace each α with γ with probability

γ α+γ and each β with δ with probability δ β+δ independently for

each occurrence.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Distribution of Parameters

We are interested in the following random variables:

1 Ak

n,α,β, the number of α’s along the

kth diagonal.

2 Bk

n,α,β, the number of β’s along the

kth diagonal.

3 X k

n,α,β, the number of non-empty

boxes along the kth diagonal.

• Pawel Hitczenko & Amanda Parshall

On the Distribution of Random Staircase Tableaux

Some Previous Results (Hitczenko-Janson)

Notation: Let αk

j and xk j denote an α or non-empty box in column

j along the kth diagonal respectively.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Some Previous Results (Hitczenko-Janson)

Notation: Let αk

j and xk j denote an α or non-empty box in column

j along the kth diagonal respectively. If a box is on the first diagonal, the distribution is: P(α1

j ) =

n − i + b n + a + b − 1.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Some Previous Results (Hitczenko-Janson)

Notation: Let αk

j and xk j denote an α or non-empty box in column

j along the kth diagonal respectively. If a box is on the first diagonal, the distribution is: P(α1

j ) =

n − i + b n + a + b − 1. If a box is not on the first diagonal, its distribution is: P(αk

j ) =

j − 1 + b (i + j + a + b − 1)(i + j + a + b − 2) P(xk

j ) =

1 (i + j + a + b − 1).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Some Previous Results Cont’d (Hitczenko-Janson)

Previous Result: A1

n,α,β − n/2

√n

d

→ N(0, 1/12)

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Some Previous Results Cont’d (Hitczenko-Janson)

Previous Result: A1

n,α,β − n/2

√n

d

→ N(0, 1/12) P(α1

j1, . . . , α1 jr ) = r

• k=1

jk − k + b n − k + a + b

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Some Previous Results Cont’d (Hitczenko-Janson)

Previous Result: A1

n,α,β − n/2

√n

d

→ N(0, 1/12) P(α1

j1, . . . , α1 jr ) = r

• k=1

jk − k + b n − k + a + b Conjecture: Ak

n,α,β and Bk n,α,β are asymptotically Poisson (k ≥ 2).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Results

Theorem Let Pois(λ) be a Poisson random variable with parameter λ. Then as n → ∞: A2

n d

→ Pois 1 2

• X 2

n d

→ Pois (1)

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Outline of Proof

Theorem (Bollob´ as) Suppose that the non-negative integer valued random variables X1, X2, ... are such that lim

n→∞ E(Xk)r → λr,

r = 0, 1, . . . Then, Xn

d

→ Pois(λ) In order to calculate the rth factorial moment, we needed to calculate first P(α2

j1, . . . , α2 jr ).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Results

Theorem Let 1 ≤ j1 < ... < jr ≤ n − 1. If jk ≤ jk+1 − 2, ∀k = 1, 2, ..., r − 1 Then, P(α2

j1, . . . , α2 jr )= r

• k=1

b + jr−k+1 − 2r + 2k − 1 (a + b + n − 2r + 2k − 1)2 . Otherwise, P(α2

j1, . . . , α2 jr ) = 0.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Proof.

Define S[i, j] to be the subtableau in Sn−i−j+2 obtained by deleting the first i − 1 rows and j − 1 columns. The following statement was proven by Hitczenko and Janson: Sn,α,β[i, j] d = Sn−i−j+2,ˆ

a,ˆ b, with ˆ

a = a + i − 1 and ˆ b = b + j − 1.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Proof.

Lemma If Sn,α,β is conditioned on Sn,α,β(n − 1, 1) = α, then the subtableau Sn,α,β[1, 3] d = Sn−2,α,β, that is L(Sn,α,β | Sn,α,β(n − 1, 1) = α) = L(Sn−2,α,β).

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

More Results

Theorem Let 1 ≤ j1 < ... < jr ≤ n − 1. If jk ≤ jk+1 − 2, ∀k = 1, 2, ..., r − 1 Then, P(x2

j1, . . . x2 jr )= r

• k=1

1 (n + a + b − r + k − 1) Otherwise, P(x2

j1, . . . , x2 jr ) = 0.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Conclusion

Current work: 3rd Diagonal and On

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Conclusion

Current work: 3rd Diagonal and On 3rd Diagonal: Preliminary calculations show A3

n,α,β is

asymptotically Poisson with parameter 1/2.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Conclusion

Current work: 3rd Diagonal and On 3rd Diagonal: Preliminary calculations show A3

n,α,β is

asymptotically Poisson with parameter 1/2. kth Diagonal: Hope to eliminate extraneous cases and get the same results.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Thank you!

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

• B. Bollob´

as. Random Graphs. Academic Press, 1985.

• R. Bundschuh.

Asymmetric exclusion process and extremal statistics of random sequences.

• Phys. Rev. E 65: 031911 2002
• S. Corteel and S. Dasse-Hartaut.

Statistics on staircase tableaux, Eulerian and Mahonian statistics. In 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO: 245-255, 2011.

• S. Corteel, R. Stanley, D. Stanton, and L. Williams.

Formulae for Askey-Wilson moments and enumeration of staircase tableaux.

• Trans. Amer. Math. Soc., 364(11):6009-6037, 2012.
• S. Corteel and L. K. Williams.

Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials.

• Proc. Natl. Acad. Sci., 107(15):6676-6730, 2010.
• S. Corteel and L. K. Williams.

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials. Duke Math. J., 159:385-415, 2011.

• S. Dasse-Hartaut and P. Hitczenko.

Greek letters in random staircase tableaux. Random Struct. Algorithms, 42:73-96, 2013.

• P. Hitczenko and S. Janson.

Weighted random staircase tableaux.

• Combin. Probab. Comput., to appear. arXiv:1212.5498
• J. MacDonald, J. Gibbs, A. Pipkin.

Kinetics of biopolymerization on nucleaic acid templates. Biopolymers, 6(1): 1-25, 1968.

• N. Rajewsky, L. Santen, A. Schadschneider, M. Schreckenberg.

The asymmetric exclusion process: Comparison of update procedures.

• J. Stat. Phys: 151-194, 1998.
• F. Spitzer.

Interaction of Markov Processes. Advances in Mathematics, 5(2): 246-290, 1970. H-T. Yau. (logt)( 2

3) law of the two dimenstional asymmetric simple exclusion process.

Annals of MathematicsSecond Series, 159(1): 337-405, 2004.

Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux