Combinatorics of exclusion processes with open boundaries Sylvie - PowerPoint PPT Presentation

Combinatorics of exclusion processes with open boundaries Sylvie Corteel (CNRS Paris 7) GGI, Florence, May 19th 2015

Koornwinder moments and the two species ASEP Sylvie Corteel (CNRS Paris 7) Lauren Williams (Berkeley) Triangular staircase tableaux Sylvie Corteel, Olya Mandelshtam (Berkeley) and Lauren Williams (Berkeley) . . .

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ C ( ⌧ ) number of tableaux of shape ⌧

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ B ( ⌧ ) /C n +1 is the probability to be in state ⌧ of the TASEP with open boundaries and N sites C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Binary trees, Paths and tableaux ⌧ 2 { � , •} N B ( ⌧ ) number of trees of canopy ⌧ M ( ⌧ ) number of paths of shape ⌧ B ( ⌧ ) /C n +1 is the probability to be in state ⌧ of the TASEP with open boundaries and N sites C ( ⌧ ) number of tableaux of shape ⌧ B ( τ ) = M ( τ ) = C ( τ ) P ⌧ C ( ⌧ ) = C n +1 Catalan numbers

Matrix Ansatz [Derrida et al 93] Matrices D and E , and vectors h W | and | V i • h W | E = h W | Z N = h W | ( D + E ) N | V i . • D | V i = | V i • DE = D + E P ( τ ) = h W | Q N i =1 [ ⌧ i D +(1 � ⌧ i ) E ] | V i Steady state ⌧ 2 { � , •} = { 0 , 1 } N . Z N

Matrix Ansatz [Derrida et al 93] Matrices D and E , and vectors h W | and | V i • h W | E = h W | Z N = h W | ( D + E ) N | V i . • D | V i = | V i • DE = D + E P ( τ ) = h W | Q N i =1 [ ⌧ i D +(1 � ⌧ i ) E ] | V i Steady state ⌧ 2 { � , •} = { 0 , 1 } N . Z N h W | = (1 , 0 , . . . ) , | V i = (1 , 0 , . . . ) T Solution: 0 1 0 1 1 1 0 0 . . . 1 0 0 0 . . . 0 1 1 0 . . . 1 1 0 0 . . . B C B C B C B C 0 0 1 1 . . . 0 1 1 0 . . . B C B C D = E = B C B C 0 0 0 1 . . . 0 0 1 1 . . . B C B C @ A @ A . . . . . . . . . . . . Motzkin paths [Zeilberger, Duchi and Schae ff er, Brak and Essam]

Matrix Ansatz [Derrida et al 93] Matrices D and E , and vectors h W | and | V i • h W | E = h W | Z N = h W | ( D + E ) N | V i . • D | V i = | V i • DE = D + E P ( τ ) = h W | Q N i =1 [ ⌧ i D +(1 � ⌧ i ) E ] | V i Steady state ⌧ 2 { � , •} = { 0 , 1 } N . Z N h W | = (1 , 0 , . . . ) , | V i = (1 , 1 , . . . ) T Solution: 0 1 0 1 0 1 0 0 . . . 1 0 0 0 . . . 0 0 1 0 . . . 1 1 0 0 . . . B C B C B C B C 0 0 0 1 . . . 1 1 1 0 . . . B C B C D = E = B C B C 0 0 0 0 . . . 1 1 1 1 . . . B C B C @ A @ A . . . . . . . . . . . . Lukasiewicz paths, Catalan tableaux

Asymmetric exclusion process with 5 parameters

Asymmetric exclusion process with 5 parameters γ α β δ q γ δ β α

Asymmetric exclusion process with 5 parameters γ α β δ q γ δ β α Matrix Ansatz • h W | ( ↵ E � � D ) = h W | • ( � D � � E ) | V i = | V i • DE = qED + D + E

Asymmetric exclusion process with 5 parameters γ α β δ q γ δ β α Matrix Ansatz • h W | ( ↵ E � � D ) = h W | � = � = 0 • ( � D � � E ) | V i = | V i • Trees ) tree like tableaux • DE = qED + D + E • Paths ) moments of AlSalam-Chihara Polynomials • Tableaux ) Permutation tableaux, Alternative tableaux

Asymmetric exclusion process with 5 parameters γ α β δ q γ δ β α Matrix Ansatz • h W | ( ↵ E � � D ) = h W | � = � = 0 • ( � D � � E ) | V i = | V i • Trees ) tree like tableaux • DE = qED + D + E • Paths ) moments of AlSalam-Chihara Polynomials • Tableaux ) Permutation tableaux, Alternative tableaux [Aval, Boussicault, C. Josuat-Verg` es, Nadeau, Viennot, Williams. . . ]

Asymmetric exclusion process with 5 parameters γ α β δ q γ δ β α General model • Moments of Askey Wilson polynomials [Uchiyama, Sasamoto, Wadati 04] • Staircase tableaux [C., Williams 10]

Askey Wilson polynomials P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b n = 1 / 2( a + 1 /a � A n � C n ) � n = A n � 1 C n / 4 A n = (1 � abq n )(1 � acq n )(1 � adq n )(1 � abcdq n − 1 ) a (1 � abcdq 2 n )(1 � abcdq 2 n − 1 ) C n = (1 � abq n − 1 )(1 � bcq n − 1 )(1 � bdq n − 1 )(1 � q n ) symmetric in a, b, c, d a (1 � abcdq 2 n − 2 )(1 � abcdq 2 n − 1 )

Askey Wilson polynomials P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b n = 1 / 2( a + 1 /a � A n � C n ) � n = A n � 1 C n / 4 A n = (1 � abq n )(1 � acq n )(1 � adq n )(1 � abcdq n − 1 ) a (1 � abcdq 2 n )(1 � abcdq 2 n − 1 ) C n = (1 � abq n − 1 )(1 � bcq n − 1 )(1 � bdq n − 1 )(1 � q n ) symmetric in a, b, c, d a (1 � abcdq 2 n − 2 )(1 � abcdq 2 n − 1 ) orthogonal ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ H z + z − 1 z + z − 1 z + z − 1 dz 4 ⇡ iz w P m P n = h n δ mn , C 2 2 2 ( z 2 ,z − 2 ; q ) ∞ ( az,a/z,bz,b/z,cz,c/z,dz,d/z ; q ) ∞ , x = ( z + z � 1 ) / 2 w ( x ) = h n = (1 � q n − 1 abcd )( q,ab,ac,ad,bc,bd,cd ; q ) n (1 � q 2 n − 1 abcd )( abcd ; q ) n

Askey Wilson polynomials P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b n = 1 / 2( a + 1 /a � A n � C n ) � n = A n � 1 C n / 4 A n = (1 � abq n )(1 � acq n )(1 � adq n )(1 � abcdq n − 1 ) a (1 � abcdq 2 n )(1 � abcdq 2 n − 1 ) C n = (1 � abq n − 1 )(1 � bcq n − 1 )(1 � bdq n − 1 )(1 � q n ) symmetric in a, b, c, d a (1 � abcdq 2 n − 2 )(1 � abcdq 2 n − 1 ) orthogonal ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ H z + z − 1 z + z − 1 z + z − 1 dz 4 ⇡ iz w P m P n = h n δ mn , C 2 2 2 ( z 2 ,z − 2 ; q ) ∞ ( az,a/z,bz,b/z,cz,c/z,dz,d/z ; q ) ∞ , x = ( z + z � 1 ) / 2 w ( x ) = h n = (1 � q n − 1 abcd )( q,ab,ac,ad,bc,bd,cd ; q ) n (1 � q 2 n − 1 abcd )( abcd ; q ) n ⇣ ⌘ ⇣ ⌘ N Moments H z + z − 1 z + z − 1 dz µ AW = 4 ⇡ iz w N C 2 2

Combinatorics of moments [Flajolet, Viennot 80s] P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b 2 � 2 � 2 b 1 � 1 � 1 (0 , 0) ( N, 0)

Combinatorics of moments [Flajolet, Viennot 80s] P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b 2 � 2 � 2 b 1 µ N = P � 1 � 1 P W ( p ) (0 , 0) ( N, 0) W ( P ) = b 1 b 2 � 1 � 2 2

Combinatorics of moments [Flajolet, Viennot 80s] P n +1 ( x ) = ( x � b n ) P n ( x ) � � n P n � 1 ( x ) b 2 � 2 � 2 b 1 µ N = P � 1 � 1 P W ( p ) (0 , 0) ( N, 0) W ( P ) = b 1 b 2 � 1 � 2 2 � 2 ( N, r ) b 1 b 1 � 1 b 0 (0 , 0) ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ N H z + z − 1 z + z − 1 z + z − 1 dz µ N,r = 4 ⇡ iz w P r C 2 2 2

Solution of the 5 parameter model [USW 04] 0 1 d \ d ] 0 · · · 0 0 d \ d ] d [ B C 0 1 1 B C d = ... B C d \ d [ 0 B C 1 2 @ A . ... ... . . q n bd d [ e [ 1 d ] n = 1 e ] n = � q n ac n = � n = (1 � q n ac )(1 � q n bd ) � n (1 � q n ac )(1 � q n bd ) � n h W | = (1 , 0 , . . . ) , | V i = (1 , 0 , . . . ) T

Solution of the 5 parameter model [USW 04] 0 1 d \ d ] 0 · · · 0 0 d \ d ] d [ B C 0 1 1 B C d = ... B C d \ d [ 0 B C 1 2 @ A . ... ... . d \ n + e \ n = b n . q n bd d [ e [ 1 d ] n = 1 e ] n = � q n ac n = � n = (1 � q n ac )(1 � q n bd ) � n (1 � q n ac )(1 � q n bd ) � n h W | = (1 , 0 , . . . ) , | V i = (1 , 0 , . . . ) T µ AW = h W | ( d + e ) N | V i N