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)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ β γ β Figure: A staircase tableau of size 7 and weight α 2 β 3 γ 3 δ 2 . Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ β Figure: A staircase tableau of size 7 and weight α 2 β 3 γ 3 δ 2 . Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. Figure: A staircase tableau of size 7 and weight α 2 β 3 γ 3 δ 2 . Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. 3 All boxes in the same column Figure: A staircase and above an α or γ are empty. tableau of size 7 and weight α 2 β 3 γ 3 δ 2 . Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Staircase Tableaux (Corteel-Williams (2009)) Definition α γ α A staircase tableau of size n is a δ Young diagram of shape β γ (n, n-1, ..., 1) such that: δ 1 The boxes are empty or contain β an α , β , γ , or δ . γ 2 Every box on the diagonal β contains a symbol. 3 All boxes in the same column Figure: A staircase and above an α or γ are empty. tableau of size 7 and 4 All boxes in the same row and to weight α 2 β 3 γ 3 δ 2 . 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 S n 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 S n to be the set of all staircase tableaux of size n . The weight of S ∈ S n is the product of all symbols in S : wt ( S ) = α N α β N β γ N γ δ N δ . Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries Cont’d As proven by Corteel and Dasse-Hartaut: Z n ( α, β, γ, δ ) := � S ∈S n wt ( S ) = n − 1 � ( α + β + δ + γ + i ( α + γ )( β + δ )) . i =0 Pawel Hitczenko & Amanda Parshall On the Distribution of Random Staircase Tableaux

Preliminaries Cont’d As proven by Corteel and Dasse-Hartaut: Z n ( α, β, γ, δ ) := � S ∈S n wt ( S ) = n − 1 � ( α + β + δ + γ + i ( α + γ )( β + δ )) . i =0 | S n | = Z n (1 , 1 , 1 , 1) = 4 n n ! 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: α ◦ A to • A : n + 1 u γ A • ◦ B to A ◦ • B : • A to A ◦ : n + 1 n + 1 q δ A ◦ • B to A • ◦ B : A ◦ to A • : n + 1 n + 1 β A • to A ◦ : 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. α γ α • ◦ for each β or γ on diagonal. δ • β γ ◦ δ • Filling rules for u ’s and q ’s: β ◦ 1 u ’s in all boxes east of a β and γ ◦ β 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 or γ . 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. α q q γ q u α • ◦ for each β or γ on diagonal. q q q q q δ • u u β u γ ◦ q q q δ • Filling rules for u ’s and q ’s: u u β ◦ 1 u ’s in all boxes east of a β and q γ ◦ β q ’s in all boxes east of a δ . ◦ 2 u ’s in all boxes north of a α or δ Figure: A staircase and q ’s in all boxes north of a β tableau and it’s type or γ . • • ◦ • ◦ ◦ ◦ 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 ) � n wt ( S ) S ∈ S ′ ′ where T is the set of all staircase tableaux of type η , S n is the set of 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