Schur processes and dimer models J er emie Bouttier Based on - PowerPoint PPT Presentation

Schur processes and dimer models J´ er´ emie Bouttier Based on joint works with Dan Betea, C´ edric Boutillier, Guillaume Chapuy, Sylvie Corteel, Sanjay Ramassamy and Mirjana Vuleti´ c Institut de Physique Th´ eorique, CEA Saclay D´ epartement de math´ ematiques et applications, ENS Paris Al´ ea 2016, March 11 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 1 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (r´ eponse: FS, VIII.23) (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (r´ eponse: FS, VIII.23) (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 Peut-on retrouver cette formule par les m´ ethodes de Kyoto? Oui, et cela se g´ en´ erale en les “pavages pentus” (arXiv:1407.0665) . 2013: Al´ ea, FPSAC, ... = ⇒ plein de nouveaux collaborateurs! J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (r´ eponse: FS, VIII.23) (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 Peut-on retrouver cette formule par les m´ ethodes de Kyoto? Oui, et cela se g´ en´ erale en les “pavages pentus” (arXiv:1407.0665) . 2013: Al´ ea, FPSAC, ... = ⇒ plein de nouveaux collaborateurs! 2014-2015: g´ en´ eration al´ eatoire (arXiv:1407.3764) , Rail Yard Graphs (arXiv:1407.3764) J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Les Al´ eas de la science 2011-2012: groupe de lecture de combinatoire au LIAFA sur les hi´ erarchies int´ egrables (travaux de l’´ ecole de Kyoto) avril 2012: visite au MSRI, partitions-pyramides (Ben Young): (1 + z 2 i − 1 ) 2 i − 1 a n z n = � � , a n ∼ ? (r´ eponse: FS, VIII.23) (1 − z 2 i ) 2 i n ≥ 0 i ≥ 1 Peut-on retrouver cette formule par les m´ ethodes de Kyoto? Oui, et cela se g´ en´ erale en les “pavages pentus” (arXiv:1407.0665) . 2013: Al´ ea, FPSAC, ... = ⇒ plein de nouveaux collaborateurs! 2014-2015: g´ en´ eration al´ eatoire (arXiv:1407.3764) , Rail Yard Graphs (arXiv:1407.3764) 2015+: extension aux cas cylindrique (p´ eriodique) et pfaffien (bords libres) (en cours avec D. Betea, P. Nejjar et M. Vuleti´ c) , etc. J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 2 / 34

Outline Introduction: bosons and fermions 1 Rail yard graphs: all Schur processes are dimer models 2 Enumeration and statistics 3 Random generation 4 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 3 / 34

Outline Introduction: bosons and fermions 1 Rail yard graphs: all Schur processes are dimer models 2 Enumeration and statistics 3 Random generation 4 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 4 / 34

Bosons, Young diagrams, integer partitions... 1 2 3 4 5 · · · 1 + 1 + 1 + 2 + 2 + 4 = 11 Integer partition: λ = (4 , 2 , 2 , 1 , 1 , 1) = (4 , 2 , 2 , 1 , 1 , 1 , 0 , 0 , . . . ), | λ | = 11. J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 5 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Each box may contain at most one particle ( • ). No particle = “hole” ( ◦ ). J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Each box may contain at most one particle ( • ). No particle = “hole” ( ◦ ). Maya diagram: there are finitely many particles on the positive side and holes on the negative side. J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Each box may contain at most one particle ( • ). No particle = “hole” ( ◦ ). Maya diagram: there are finitely many particles on the positive side and holes on the negative side. Vacuum: − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Each box may contain at most one particle ( • ). No particle = “hole” ( ◦ ). Maya diagram: there are finitely many particles on the positive side and holes on the negative side. Vacuum: − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Any other diagram is obtained by a finite number of operations: adding a particle with positive energy removing a particle with negative energy J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Fermions and Maya diagrams Boxes labeled by half-integers (“energy levels”, positive or negative): − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Each box may contain at most one particle ( • ). No particle = “hole” ( ◦ ). Maya diagram: there are finitely many particles on the positive side and holes on the negative side. Vacuum: − 9 − 7 − 5 − 3 − 1 1 3 5 7 9 · · · · · · 2 2 2 2 2 2 2 2 2 2 Any other diagram is obtained by a finite number of operations: adding a particle with positive energy removing a particle with negative energy (total energy increases in both cases!) J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 6 / 34

Boson-fermion correspondence: combinatorial version Maya diagrams are in bijection with pairs ( λ, c ) with λ a partition and c an integer (the charge). Here λ = (4 , 2 , 1). J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 7 / 34

Boson-fermion correspondence: combinatorial version Maya diagrams are in bijection with pairs ( λ, c ) with λ a partition and c an integer (the charge). Here λ = (4 , 2 , 1). For i ≥ 1, the i -th rightmost particle is at position λ i − i + c + 1 / 2 and the i -th leftmost hole is at position − λ ′ i + i + c − 1 / 2 ( λ ′ : conjugate partition). J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 7 / 34

Boson-fermion correspondence: combinatorial version Maya diagrams are in bijection with pairs ( λ, c ) with λ a partition and c an integer (the charge). Here λ = (4 , 2 , 1). For i ≥ 1, the i -th rightmost particle is at position λ i − i + c + 1 / 2 and the i -th leftmost hole is at position − λ ′ i + i + c − 1 / 2 ( λ ′ : conjugate partition). Total energy is | λ | + c 2 / 2. J´ er´ emie Bouttier (IPhT/DMA) Schur processes and dimer models 11 March 2016 7 / 34