The combinatorics of some exclusion model in physics
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1 The combinatorics of some exclusion model in physics Orsay 5 April 2012 Xavier G. Viennot CNRS, Bordeaux, France
2 The PASEP Partialy asymmetric exclusion process
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4
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7 The Matrix Ansatz Derrida, Evans, Hakim, Pasquier (1993)
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9 q=0 TASEP
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12 transitions de phase
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14 Combinatorics of the TASEP Shapiro, Zeilberger, 1982
15 Combinatorics of the TASEP Shapiro, Zeilberger, 1982
16 TASEP Shapiro, Zeilberger (1982) Brak, Essam (2003), Duchi, Schaeffer, (2004), Angel (2005), XGV (2007) (P) ASEP Brak, Corteel, Essam, Parviainen, Rechnitzer (2006) Corteel, Williams (2006,..., 2010) Corteel, Stanton, Stanley, Williams (2011) Josuat-Vergès (2008,..., 2010) Derrida,... Mallick,... Golinelli, Mallick (2006)...
17 The PASEP algebra DE = qed + E +D
18 D D E D E E D E D DE (D E) E D E DDE(E)EDE + DDE(ED)EDE + DDE(D)EDE
19 I D I I I E I I D E I I I
20 D E
21 D E
22 D E
23 D E
24 D E
25 D E
26 D E
27 D E
28 D E
29 D E
30 D E
31 D E
32 D E
33 D E
34 D E
35 D E
36 D E
37 D E
38 D E
39 D E
40 E D E E E D
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42
43
44
45 alternative tableau Ferrers diagram (= Young diagram)
46 alternative tableau
47 alternative tableau n = 12
48 E D E E E D
49
50 rows columns blue red
51 "normal ordering" Heisenberg operators U, D UD = DU + I
52 Lemma Every word w with letters U and D can be written in a unique way
53 U D = D U U U D D U U D U U Towers placements on a Ferrers diagram
54
55 U D permutations n!
56 stationary probabilities for the PASEP
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58 permutation tableau
59 M. Josuat-Vergès (2007)
60 permutation tableaux
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62
63
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65 TASEP Totally asymmetric exclusion process
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68 Bijection alternative Catalan tableaux binary trees
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84
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86 Derrida, Evans, Hakim, Pasquier (1993)
87 combinatorial physics or integrable combinatorics
88 number of alternative tableaux
89 alternative tableau n = 12
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91 The exchange-fusion algorithm
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93
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95
96 (43) 2 8
97 (43) 8 7
98 9 6 2 (43)
99 (43) 8 1 5
100 6 2 (43)
101 6 2 (43) (89) 7 1 5
102 6 (43) 2 (89) 7 1 5
103 6 7 (89) (432) 1 5
104 6 1 7 (432) (89) 5
105 6 7 (432) 1 (89) 5
106 6 7 (89) (4321) 5
107 6 7 (89) (4321) 5
108 6 (789) (4321) 5
109 exchangefusion algorithm (789) (4321) 5 6
110 The inverse exchangefusion algorithm
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112
113 7,8,9 1,2,3,4 5 6
114 7 8,9 1,2,3,4 5 6
115 ,2,3,4 5 6
116 ,3,
117 ,
118
119
120 The "cellular Ansatz" representation of the operators E and D DE = ED + E + D
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122 S K S = +
123 S S S S S
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125
126
127 The "cellular Ansatz" "planar construction" of a bijection permutations alternative tableaux
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129
130 A 1 2 K 3 A 4 5 S S J A S
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132
133 A 1 2 K 3 A 4 5 S S J A S
134 A K K A A S S J A S
135 A K K A A S A S S J A S
136 A K K A A I J S A S S J A S
137 A K K A A I J S A S K S I J A S
138 A K K A A I J S A S K K S J I J A S
139 A K K A A I J S A S K K S J I J A K S I
140 A K K A A I J S A S K K S J I J A K K S J I
141 A K K A A I J S A S K K S J I J A K K K S I J I
142 A K K A A I J S A S K K S J I J A K K K K S J I J I
143
144 Laguerre histories combinatorial theory of orthogonal polynomials The FV bijection Françon-XV 1978
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151 (formal) orthogonal polynomials
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154 combinatorial interpretation of the moments
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159 continued fractions 1980 orthogonal polynomials P. Flajolet Lecture Note X.V. 1983
160 Laguerre histories and Laguerre polynomials The FV bijection Françon-XV 1978
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166 3 parameters
167 permutation tableau
168 E D E E E D
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170 xy(x+y)(x+1+y)...(x+n-2+y)
171 q-analogue of Laguerre histories choices function q-laguerre : q 4
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173 quadratic algebra operators data structures and orthogonal polynomials
174 Primitive operations for dictionnaries data structure: add or delete any elements, asking questions (with positive or negative answer)
175 number of choices for each primitive operations
176 this corresponds to the n! restricted Laguerre histories this valuation corresponds to the (n+1)! enlarged Laguerre histories
177 priority queue Polya urn
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181 A S - S A = I
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183 The cellular Ansatz From quadratic algebra Q to combinatorial objects (Q-tableaux) and bijections
184 The Robinson-Schensted correspondence between permutations and pair of (standard) Young tableaux with the same shape
185 Operators U and D U D adding or deleting a cell in a Ferrers diagram Young lattice UD = DU + I
186 U D I I
187 "The cellular Ansatz" Physics "normal ordering" UD = DU + Id Weyl-Heisenberg DE = qed + E + D PASEP quadratic algebra Q commutations rewriting rules planarisation combinatorial objects on a 2d lattice towers placements permutations tableaux alternatifs Q-tableaux ex: ASM, (alternating sign matrices) FPL(fully packed loops) tilings, 8-vertex planar automata bijections RSK representation by operators data structures "histories" orthogonal polynomials pairs of Tableaux Young permutations Laguerre histories? Koszul algebras duality
188 ASM Alternating sign matrices
189
190 random FPL
191 Razumov - Stroganov (ex)- conjecture proof by : L. Cantini and A.Sportiello (March 2010) arxiv: [math.c0] completely combinatorial proof
192 Around the Razumov-Stroganov conjecture Philippe Di Francesco, Paul Zinn-Justin ( ) De Gier, Pyatov (2007) Knizhnik - Zamolodchikov equation qkz TSSCPP ASM
193 The 8-vertex algebra (or XYZ - algebra) (or Z - algebra)
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199 2
200 2 n(n-1)/2 A (2) n Elkies, Kuperberg, Larsen, Propp (1992)
201 random Aztec tilings
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204 example: binomial determinant
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206 general PASEP
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208 Askey-Wilson integral
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213 Askey tableau
214 Askey-Wilson
215 Askey-Wilson
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217 staircase tableaux Corteel, Williams, 2009 Corteel, Stanley, Stanton, Williams, 2010
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224 relation with moments of Askey-Wilson polynomials Corteel, Williams, 2009 Corteel, Stanley, Stanton, Williams, 2010
225 Random Combinatorics matrix theory orthogonal polynomials tridiagonal matrices PASEP physics
226 "The cellular Ansatz" Physics "normal ordering" UD = DU + Id Weyl-Heisenberg DE = qed + E + D PASEP quadratic algebra Q commutations rewriting rules planarisation combinatorial objects on a 2d lattice towers placements permutations tableaux alternatifs Q-tableaux ex: ASM, (alternating sign matrices) FPL(fully packed loops) tilings, 8-vertex planar automata bijections RSK representation by operators data structures "histories" orthogonal polynomials pairs of Tableaux Young permutations Laguerre histories? Koszul algebras duality
227 Thank you!
228 pour plus de détails voir les diaporamas du cours donné à Talca: Cours XGV, Universidad de Talca (December January 2011) Combinatorics and interactions (with physics) (24h) «The Cellular Ansatz» accessible sur les sites: Recherche, cv, publications, exposés, diaporamas, livres, petite école, photos: voir ma page personnelle ici Vulgarisation scientifique voir la page de l'association Cont'Science
229 Ch 0 Introduction Ch 1 Ordinary generating function, the Catalan garden Ch1a (1/12/2010, 54 p.) Ch 1b (7/12/2010, 81 p.) Ch 1c (7/12/2010, 30 p.) algebraic complements in relation with physics Ch 2 Exponential generating functions, permutations Ch 2a (22/12/2010, 40 p.) Ch 2b (4/01/2010, 63 p.) Ch 2c (4/01/2010, 33 p.) Permutations: Laguerre histories Cours XGV Universidad de Talca (December January 2011) 24 h Combinatorics and interactions (with physics) «The Cellular Ansatz» Ch 3 Permutations and Young tableaux, the Robinson-Schensted correspondence (RSK) Ch 3a (6/01/2011, 117 p.) Ch 3b (6, 11/01/2011, 121 p.) RSK and operators Ch 4 Alternative tableaux and the PASEP (partially asymmetric exclusion process) Ch 4a (13/01/2011, 98p.) Ch 4b (13, 18/01/2011, 102 p.) alternative tableaux and the PASEP Ch 4c (18/01/2011, 81 p.) complements Ch 5 Combinatorial theory of orthogonal polynomials (20/01/2011, 110 p.) Ch 6 "jeu de taquin" for binary trees, Catalan tableaux and the TASEP Ch 6a (24/01/2011, 98 p.) Ch 6b (24/01/2011, 111 p.) alternative tableaux and increasing/alternative binary trees Ch 6c (24/01/2011, 21 p.) Catalan tableaux and the Loday-Ronco algebra Ch 7 The cellular Ansatz Ch 7a (25/01/2011, 117 p.) Ch 7b (25/01/2011, 49 p.) complements
230 PASEP Matrix Ansatz PASEP algebra AT "normal ordering" stationary probabilities permutation tableaux TASEP Bijection CAT - BT The exchange-fusion algorithm representation E, D The "cellular Ansatz" FV bijection formal OP Laguerre polynomials 3 parameters data structures cellular Ansatz Q ASM R-S 8-vertex algebra general PASEP A-W integral Askey tableau staircase tableaux
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