integrable discrete time dynamics generalized exclusion
play

Integrable discrete-time dynamics, generalized exclusion processes - PowerPoint PPT Presentation

Jul 14, 2023 •163 likes •1.41k views

Physical motivations and framework Integrable discrete-time dynamics Generalized exclusion processes Integrable discrete-time dynamics, generalized exclusion processes and "fused" matrix ansatz Matthieu VANICAT University of

  1. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes The probabilities are encompassed in a vector |P t � = � C P t ( C ) |C� . Master equation |P t + 1 � = M |P t � , where M C ′ , C = m ( C → C ′ ) is a discrete-time Markov matrix the entries are non-negative. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  2. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes The probabilities are encompassed in a vector |P t � = � C P t ( C ) |C� . Master equation |P t + 1 � = M |P t � , where M C ′ , C = m ( C → C ′ ) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  3. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes The probabilities are encompassed in a vector |P t � = � C P t ( C ) |C� . Master equation |P t + 1 � = M |P t � , where M C ′ , C = m ( C → C ′ ) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one. The stationary state |S� = � C S ( C ) |C� satisfies |S� = M |S� . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  4. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes The probabilities are encompassed in a vector |P t � = � C P t ( C ) |C� . Master equation |P t + 1 � = M |P t � , where M C ′ , C = m ( C → C ′ ) is a discrete-time Markov matrix the entries are non-negative. the sum over each column is one. The stationary state |S� = � C S ( C ) |C� satisfies |S� = M |S� . In this talk: integrability of M means M = t ( κ ) , with [ t ( x ) , t ( y )] = 0 , ∀ x , y . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  5. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  6. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  7. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  8. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  9. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  10. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries A configuration is given by τ = ( τ 1 , τ 2 , . . . , τ L ) Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  11. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes (Generalized) exclusion processes One-dimensional lattice with L sites Maximal number s of particles on each site The particles evolve randomly on the lattice The system is coupled to particle reservoirs at the boundaries A configuration is given by τ = ( τ 1 , τ 2 , . . . , τ L ) τ i is the number of particles on site i . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  12. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update α β γ δ The stochastic dynamics is defined in two steps: Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  13. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  14. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  15. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  16. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  17. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ During the second step, neighboring even-odd sites are paired Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  18. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p p p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ During the second step, neighboring even-odd sites are paired In the bulk, particles jump if possible to the left/right with proba p Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  19. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p p p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ During the second step, neighboring even-odd sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  20. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes Example: SSEP with sublattice parallel update p p p p p α β γ δ The stochastic dynamics is defined in two steps: During the first step, neighboring odd-even sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the right boundary, particles leave with proba β or enter with δ During the second step, neighboring even-odd sites are paired In the bulk, particles jump if possible to the left/right with proba p Exclusion principle At the left boundary, particles enter with proba α or leave with γ Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  21. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes p p β δ Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  22. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes p p β δ The associated discrete-time Markov matrix reads M = U e U o Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  23. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes p p β δ The associated discrete-time Markov matrix reads M = U e U o with operators U e and U o defined as L − 1 L − 1 � � 2 2 U o = U e = B 1 U 2 k − 1 , 2 k B L and U 2 k , 2 k + 1 . k = 1 k = 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  24. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes p p β δ The associated discrete-time Markov matrix reads M = U e U o with operators U e and U o defined as L − 1 L − 1 � � 2 2 U o = U e = B 1 U 2 k − 1 , 2 k B L and U 2 k , 2 k + 1 . k = 1 k = 1 | 0 � ⊗ | 0 � | 0 � ⊗ | 1 � | 1 � ⊗ | 0 � | 1 � ⊗ | 1 �   | 0 � | 1 � | 0 � | 1 � | 0 � ⊗ | 0 � 1 0 0 0 � � � � | 0 � | 0 � 1 − α γ  0 1 − p p 0  | 0 � ⊗ | 1 � 1 − δ β B = U = B =   α 1 − γ 0 p 1 − p 0 δ 1 − β | 1 � | 1 � | 1 � ⊗ | 0 � 0 0 0 1 | 1 � ⊗ | 1 � Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  25. Physical motivations and framework Markov dynamics and non-equilibrium stationary state Integrable discrete-time dynamics Exclusion processes Generalized exclusion processes U e B U U U o U U B U e B U U 1 2 3 4 5 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  26. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Integrable discrete-time dynamics Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  27. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  28. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? The building block is a matrix ˇ R ( z ) ∈ End ( V ⊗ V ) Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  29. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? The building block is a matrix ˇ R ( z ) ∈ End ( V ⊗ V ) Yang-Baxter equation R 23 ( z 1 − z 2 ) ˇ ˇ R 12 ( z 1 − z 3 ) ˇ R 23 ( z 2 − z 3 ) = ˇ R 12 ( z 2 − z 3 ) ˇ R 23 ( z 1 − z 3 ) ˇ R 12 ( z 1 − z 2 ) . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  30. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? The building block is a matrix ˇ R ( z ) ∈ End ( V ⊗ V ) Yang-Baxter equation R 23 ( z 1 − z 2 ) ˇ ˇ R 12 ( z 1 − z 3 ) ˇ R 23 ( z 2 − z 3 ) = ˇ R 12 ( z 2 − z 3 ) ˇ R 23 ( z 1 − z 3 ) ˇ R 12 ( z 1 − z 2 ) . We also require Unitarity: ˇ R ( z )ˇ R ( − z ) = 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  31. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? The building block is a matrix ˇ R ( z ) ∈ End ( V ⊗ V ) Yang-Baxter equation R 23 ( z 1 − z 2 ) ˇ ˇ R 12 ( z 1 − z 3 ) ˇ R 23 ( z 2 − z 3 ) = ˇ R 12 ( z 2 − z 3 ) ˇ R 23 ( z 1 − z 3 ) ˇ R 12 ( z 1 − z 2 ) . We also require Unitarity: ˇ R ( z )ˇ R ( − z ) = 1 Markov property: � 1 | ⊗ � 1 | ˇ R ( z ) = � 1 | ⊗ � 1 | Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  32. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form How to construct an integrable Markov matrix M ? The building block is a matrix ˇ R ( z ) ∈ End ( V ⊗ V ) Yang-Baxter equation R 23 ( z 1 − z 2 ) ˇ ˇ R 12 ( z 1 − z 3 ) ˇ R 23 ( z 2 − z 3 ) = ˇ R 12 ( z 2 − z 3 ) ˇ R 23 ( z 1 − z 3 ) ˇ R 12 ( z 1 − z 2 ) . We also require Unitarity: ˇ R ( z )ˇ R ( − z ) = 1 Markov property: � 1 | ⊗ � 1 | ˇ R ( z ) = � 1 | ⊗ � 1 | Example: SSEP ( s = 1 , V = C 2 )   1 0 0 0   1 z R ( z ) = 1 + zP 0 0   ˇ 1 + z 1 + z 1 + z =   z 1 0 0   1 + z 1 + z 0 0 0 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  33. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form The integrable boundary conditions are given by K ( z ) , K ( z ) ∈ End ( V ) Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  34. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form The integrable boundary conditions are given by K ( z ) , K ( z ) ∈ End ( V ) Reflection equation R ( z 1 − z 2 ) K 1 ( z 1 )ˇ ˇ R ( z 1 + z 2 ) K 1 ( z 2 ) = K 1 ( z 2 )ˇ R ( z 1 + z 2 ) K 1 ( z 1 )ˇ R ( z 1 − z 2 ) . and a similar reflection equation for K ( z ) . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  35. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form The integrable boundary conditions are given by K ( z ) , K ( z ) ∈ End ( V ) Reflection equation R ( z 1 − z 2 ) K 1 ( z 1 )ˇ ˇ R ( z 1 + z 2 ) K 1 ( z 2 ) = K 1 ( z 2 )ˇ R ( z 1 + z 2 ) K 1 ( z 1 )ˇ R ( z 1 − z 2 ) . and a similar reflection equation for K ( z ) . We also require Unitarity: K ( z ) K ( − z ) = 1, K ( z ) K ( − z ) = 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  36. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form The integrable boundary conditions are given by K ( z ) , K ( z ) ∈ End ( V ) Reflection equation R ( z 1 − z 2 ) K 1 ( z 1 )ˇ ˇ R ( z 1 + z 2 ) K 1 ( z 2 ) = K 1 ( z 2 )ˇ R ( z 1 + z 2 ) K 1 ( z 1 )ˇ R ( z 1 − z 2 ) . and a similar reflection equation for K ( z ) . We also require Unitarity: K ( z ) K ( − z ) = 1, K ( z ) K ( − z ) = 1 Markov property: � 1 | K ( z ) = � 1 | , � 1 | K ( z ) = � 1 | Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  37. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form The integrable boundary conditions are given by K ( z ) , K ( z ) ∈ End ( V ) Reflection equation R ( z 1 − z 2 ) K 1 ( z 1 )ˇ ˇ R ( z 1 + z 2 ) K 1 ( z 2 ) = K 1 ( z 2 )ˇ R ( z 1 + z 2 ) K 1 ( z 1 )ˇ R ( z 1 − z 2 ) . and a similar reflection equation for K ( z ) . We also require Unitarity: K ( z ) K ( − z ) = 1, K ( z ) K ( − z ) = 1 Markov property: � 1 | K ( z ) = � 1 | , � 1 | K ( z ) = � 1 | Example: SSEP ( s = 1 , V = C 2 )     ( c − a ) z + 1 ( b − d ) z − 1 2 cz 2 bz ( a + c ) z + 1 ( a + c ) z + 1 ( b + d ) z − 1 ( b + d ) z − 1     . K ( z ) = and K ( z ) = ( a − c ) z + 1 ( d − b ) z − 1 2 az 2 dz ( a + c ) z + 1 ( a + c ) z + 1 ( b + d ) z − 1 ( b + d ) z − 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  38. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  39. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  40. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Main property [ t ( x | z ) , t ( y | z )] = 0 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  41. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Main property [ t ( x | z ) , t ( y | z )] = 0 Remarks: z = z 1 , z 2 , . . . , z L are the inhomogeneity parameters. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  42. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Main property [ t ( x | z ) , t ( y | z )] = 0 Remarks: z = z 1 , z 2 , . . . , z L are the inhomogeneity parameters. Usually one put z 1 = z 2 = · · · = z L = 0 to get Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  43. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Main property [ t ( x | z ) , t ( y | z )] = 0 Remarks: z = z 1 , z 2 , . . . , z L are the inhomogeneity parameters. Usually one put z 1 = z 2 = · · · = z L = 0 to get L − 1 � ′ t ′ ( 0 ) = K ′ R ′ ˇ 1 ( 0 ) + 2 k , k + 1 ( 0 ) + K L ( 0 ) k = 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  44. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous transfer matrix � � � t ( z | z ) = tr 0 K 0 ( z ) R 0 L ( z − z L ) . . . R 01 ( z − z 1 ) K 0 ( z ) R 10 ( z + z 1 ) . . . R L 0 ( z + z L ) � � with R ( z ) = P . ˇ � R ( z ) and K ( z ) = tr 0 K 0 ( − z ) R 01 ( − 2 z ) P 01 . Main property [ t ( x | z ) , t ( y | z )] = 0 Remarks: z = z 1 , z 2 , . . . , z L are the inhomogeneity parameters. Usually one put z 1 = z 2 = · · · = z L = 0 to get L − 1 � ′ t ′ ( 0 ) = K ′ R ′ ˇ 1 ( 0 ) + 2 k , k + 1 ( 0 ) + K L ( 0 ) k = 1 t ′ ( 0 ) is then interpreted as a continuous-time Markov matrix or as a quantum Hamiltonian. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  45. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form General method to get discrete-time process from transfer matrix Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  46. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z 1 = z 3 = z 5 = · · · = z L = κ and z 2 = z 4 = z 6 = · · · = z L − 1 = − κ Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  47. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z 1 = z 3 = z 5 = · · · = z L = κ and z 2 = z 4 = z 6 = · · · = z L − 1 = − κ A direct computation yields K 1 ( κ )ˇ R 23 ( 2 κ )ˇ R 45 ( 2 κ ) . . . ˇ t ( κ ) = R L − 1 , L ( 2 κ ) × ˇ R 12 ( 2 κ )ˇ R 34 ( 2 κ ) . . . ˇ R L − 2 , L − 1 ( 2 κ ) K L ( − κ ) , Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  48. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z 1 = z 3 = z 5 = · · · = z L = κ and z 2 = z 4 = z 6 = · · · = z L − 1 = − κ A direct computation yields K 1 ( κ )ˇ R 23 ( 2 κ )ˇ R 45 ( 2 κ ) . . . ˇ t ( κ ) = R L − 1 , L ( 2 κ ) × ˇ R 12 ( 2 κ )ˇ R 34 ( 2 κ ) . . . ˇ R L − 2 , L − 1 ( 2 κ ) K L ( − κ ) , We can define a discrete-time Markov matrix as M = t ( κ ) = U e U o Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  49. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form General method to get discrete-time process from transfer matrix We choose staggered inhomogeneity parameters z 1 = z 3 = z 5 = · · · = z L = κ and z 2 = z 4 = z 6 = · · · = z L − 1 = − κ A direct computation yields K 1 ( κ )ˇ R 23 ( 2 κ )ˇ R 45 ( 2 κ ) . . . ˇ t ( κ ) = R L − 1 , L ( 2 κ ) × ˇ R 12 ( 2 κ )ˇ R 34 ( 2 κ ) . . . ˇ R L − 2 , L − 1 ( 2 κ ) K L ( − κ ) , We can define a discrete-time Markov matrix as M = t ( κ ) = U e U o with operators U e and U o defined as L − 1 L − 1 � � 2 2 U o = U e = B 1 U 2 k − 1 , 2 k B L and U 2 k , 2 k + 1 . k = 1 k = 1 where U = ˇ R ( 2 κ ) , B = K ( κ ) and B = K ( − κ ) . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  50. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form U e B U U U o U U B U e B U U 1 2 3 4 5 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  51. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  52. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? We construct it in matrix product form . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  53. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? We construct it in matrix product form . The building block is a vector A ( z ) with non-commuting entries. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  54. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? We construct it in matrix product form . The building block is a vector A ( z ) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R ( z 1 − z 2 ) A ( z 1 ) ⊗ A ( z 2 ) = A ( z 2 ) ⊗ A ( z 1 ) Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  55. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? We construct it in matrix product form . The building block is a vector A ( z ) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R ( z 1 − z 2 ) A ( z 1 ) ⊗ A ( z 2 ) = A ( z 2 ) ⊗ A ( z 1 ) Example: SSEP ( s = 1 , V = C 2 ) � � − z + E A ( z ) = z + D Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  56. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form What is the associated stationary state? We construct it in matrix product form . The building block is a vector A ( z ) with non-commuting entries. The commutation relations between the entries are given by Zamolodchikov-Faddeev (ZF) relation ˇ R ( z 1 − z 2 ) A ( z 1 ) ⊗ A ( z 2 ) = A ( z 2 ) ⊗ A ( z 1 ) Example: SSEP ( s = 1 , V = C 2 ) � � − z + E A ( z ) = z + D The ZF relation implies DE − ED = D + E . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  57. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form We also need two boundary vectors � � W | and | V � � Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  58. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form We also need two boundary vectors � � W | and | V � � The algebraic relations between the vectors and the entries of A ( z ) are given by Ghoshal-Zamolodchikov (GZ) relations � = ¯ � � W | A ( z ) = � � W | K ( z ) A ( − z ) , A ( z ) | V � K ( z ) A ( − z ) | V � � . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  59. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form We also need two boundary vectors � � W | and | V � � The algebraic relations between the vectors and the entries of A ( z ) are given by Ghoshal-Zamolodchikov (GZ) relations � = ¯ � � W | A ( z ) = � � W | K ( z ) A ( − z ) , A ( z ) | V � K ( z ) A ( − z ) | V � � . Example: SSEP ( s = 1 , V = C 2 ) The GZ relations imply � � � � � � W | a E − c D − 1 = 0 , and b D − d E − 1 | V � � = 0 . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  60. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous ground-state |S ( z 1 , z 2 , . . . , z L ) � = 1 � � W | A ( z 1 ) ⊗ A ( z 2 ) ⊗ · · · ⊗ A ( z L ) | V � � , Z L Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  61. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous ground-state |S ( z 1 , z 2 , . . . , z L ) � = 1 � � W | A ( z 1 ) ⊗ A ( z 2 ) ⊗ · · · ⊗ A ( z L ) | V � � , Z L Using ZF and GZ relations we can show t ( z i | z ) |S ( z 1 , z 2 , . . . , z L ) � = |S ( z 1 , z 2 , . . . , z L ) � , for 1 ≤ i ≤ L , Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  62. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous ground-state |S ( z 1 , z 2 , . . . , z L ) � = 1 � � W | A ( z 1 ) ⊗ A ( z 2 ) ⊗ · · · ⊗ A ( z L ) | V � � , Z L Using ZF and GZ relations we can show t ( z i | z ) |S ( z 1 , z 2 , . . . , z L ) � = |S ( z 1 , z 2 , . . . , z L ) � , for 1 ≤ i ≤ L , Hence the steady state of the discrete-time process is |S� = 1 � � W | A ( κ ) ⊗ A ( − κ ) ⊗ A ( κ ) ⊗ · · · ⊗ A ( κ ) | V � � . Z L Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  63. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Inhomogeneous ground-state |S ( z 1 , z 2 , . . . , z L ) � = 1 � � W | A ( z 1 ) ⊗ A ( z 2 ) ⊗ · · · ⊗ A ( z L ) | V � � , Z L Using ZF and GZ relations we can show t ( z i | z ) |S ( z 1 , z 2 , . . . , z L ) � = |S ( z 1 , z 2 , . . . , z L ) � , for 1 ≤ i ≤ L , Hence the steady state of the discrete-time process is |S� = 1 � � W | A ( κ ) ⊗ A ( − κ ) ⊗ A ( κ ) ⊗ · · · ⊗ A ( κ ) | V � � . Z L We have indeed M |S� = |S� . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  64. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Example: SSEP with sublattice parallel update � � � � � � |S� = 1 − κ + E κ + E − κ + E Z L � � W | ⊗ ⊗ · · · ⊗ | V � � . κ + D − κ + D κ + D Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  65. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Example: SSEP with sublattice parallel update � � � � � � |S� = 1 − κ + E κ + E − κ + E Z L � � W | ⊗ ⊗ · · · ⊗ | V � � . κ + D − κ + D κ + D S ( 1 , 1 , 0 , 1 , 0 ) = � � W | ( κ + D )( − κ + D )( − κ + E )( − κ + D )( − κ + E ) | V � � Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  66. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Example: SSEP with sublattice parallel update � � � � � � |S� = 1 − κ + E κ + E − κ + E Z L � � W | ⊗ ⊗ · · · ⊗ | V � � . κ + D − κ + D κ + D S ( 1 , 1 , 0 , 1 , 0 ) = � � W | ( κ + D )( − κ + D )( − κ + E )( − κ + D )( − κ + E ) | V � � Normalization Γ � � 1 1 � = ( a + c ) L ( b + d ) L L + a + c + � W | ( E + D ) L | V � b + d Γ � � Z L = � ( ab − cd ) L 1 1 a + c + b + d Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  67. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Example: SSEP with sublattice parallel update � � � � � � |S� = 1 − κ + E κ + E − κ + E Z L � � W | ⊗ ⊗ · · · ⊗ | V � � . κ + D − κ + D κ + D S ( 1 , 1 , 0 , 1 , 0 ) = � � W | ( κ + D )( − κ + D )( − κ + E )( − κ + D )( − κ + E ) | V � � Normalization Γ � � 1 1 � = ( a + c ) L ( b + d ) L L + a + c + � W | ( E + D ) L | V � b + d Γ � � Z L = � ( ab − cd ) L 1 1 a + c + b + d Mean particle density � b + d − i � � � a 1 d 1 L + + i − 1 + a + c b + d a + c � τ i � = , 1 1 L + a + c + b + d − 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  68. Physical motivations and framework Integrability and transfer matrix Integrable discrete-time dynamics Integrable discrete-time dynamics Generalized exclusion processes Stationary state in matrix product form Example: SSEP with sublattice parallel update � � � � � � |S� = 1 − κ + E κ + E − κ + E Z L � � W | ⊗ ⊗ · · · ⊗ | V � � . κ + D − κ + D κ + D S ( 1 , 1 , 0 , 1 , 0 ) = � � W | ( κ + D )( − κ + D )( − κ + E )( − κ + D )( − κ + E ) | V � � Normalization Γ � � 1 1 � = ( a + c ) L ( b + d ) L L + a + c + � W | ( E + D ) L | V � b + d Γ � � Z L = � ( ab − cd ) L 1 1 a + c + b + d Mean particle density � b + d − i � � � a 1 d 1 L + + i − 1 + a + c b + d a + c � τ i � = , 1 1 L + a + c + b + d − 1 Mean particle current a d a + c − b + d � J � = 2 κ b + d − 1 . 1 1 L + a + c + Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  69. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Generalized exclusion processes Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  70. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  71. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on C s + 1 ⊗ C s + 1 instead of C 2 ⊗ C 2 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  72. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on C s + 1 ⊗ C s + 1 instead of C 2 ⊗ C 2 Such a R matrix can be obtain through fusion procedure Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  73. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on C s + 1 ⊗ C s + 1 instead of C 2 ⊗ C 2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R -matrix associated to the SSEP: spin 1 / 2 representation of the universal R -matrix of Y (ˆ sl 2 ) . Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  74. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on C s + 1 ⊗ C s + 1 instead of C 2 ⊗ C 2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R -matrix associated to the SSEP: spin 1 / 2 representation of the universal R -matrix of Y (ˆ sl 2 ) . Generalized exclusion processes: higher spin representations of the universal R -matrix. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  75. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Physical motivation: construct integrable processes where the exclusion constraint is relaxed: s > 1 instead of s = 1. Procedure: use the integrable sublattice parallel update with R matrix acting on C s + 1 ⊗ C s + 1 instead of C 2 ⊗ C 2 Such a R matrix can be obtain through fusion procedure Fusion procedure in a nutshell R -matrix associated to the SSEP: spin 1 / 2 representation of the universal R -matrix of Y (ˆ sl 2 ) . Generalized exclusion processes: higher spin representations of the universal R -matrix. Constructed from spin 1 / 2 R -matrix: we take tensor products of spin 1 / 2 representation and project on the appropriate invariant subspace. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  76. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Example: Generalized SSEP Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  77. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Example: Generalized SSEP Consider the R -matrix R ( z ) of the SSEP and the projectors   � 1 � 1 0 0 0 0 0  0 1 / 2 0  Q ( l ) = Q ( r ) = 0 1 1 0 and   0 1 / 2 0 0 0 0 1 0 0 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  78. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Example: Generalized SSEP Consider the R -matrix R ( z ) of the SSEP and the projectors   � 1 � 1 0 0 0 0 0  0 1 / 2 0  Q ( l ) = Q ( r ) = 0 1 1 0 and   0 1 / 2 0 0 0 0 1 0 0 1 Fusion of the “second tensor space” � � � � z − 1 z + 1 R i ,< jk > ( z ) = Q ( l ) Q ( r ) jk R ij R ik jk . 2 2 R i ,< jk > ( z ) acts on C 2 ⊗ C 3 and satisfies some Yang-Baxter equation. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  79. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz Example: Generalized SSEP Consider the R -matrix R ( z ) of the SSEP and the projectors   � 1 � 1 0 0 0 0 0  0 1 / 2 0  Q ( l ) = Q ( r ) = 0 1 1 0 and   0 1 / 2 0 0 0 0 1 0 0 1 Fusion of the “second tensor space” � � � � z − 1 z + 1 R i ,< jk > ( z ) = Q ( l ) Q ( r ) jk R ij R ik jk . 2 2 R i ,< jk > ( z ) acts on C 2 ⊗ C 3 and satisfies some Yang-Baxter equation. Fusion of the “first tensor space” � � � � z + 1 z − 1 R ( z ) = R < hi >,< jk > ( z ) = Q ( l ) Q ( r ) hi . hi R h ,< jk > R i ,< jk > 2 2 R ( z ) acts on C 3 ⊗ C 3 and satisfies the Yang-Baxter equation, Markov property and unitarity. Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  80. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz We have explicitly   1 0 0 0 0 0 0 0 0   z 2 0 0 0 0 0 0 0   z + 2 z + 2   z ( z − 1 ) z 2 0 0 0 0 0 0   ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 )   2 z  0 0 0 0 0 0 0   z + 2 z + 2  z 2 + z + 2   4 z 4 z R ( z ) = 0 0 0 0 0 0   ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 )   z 2 0 0 0 0 0 0 0   z + 2 z + 2   z ( z − 1 ) 2 z  0 0 0 0 0 0   ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 ) ( z + 1 )( z + 2 )   2  z 0 0 0 0 0 0 0 z + 2 z + 2 0 0 0 0 0 0 0 0 1 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  81. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz A similar fusion procedure can also be applied to the K -matrices Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  82. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz A similar fusion procedure can also be applied to the K -matrices � � � � z − 1 z + 1 K < ij > ( z ) = Q ( l ) Q ( r ) K ( z ) = ij K i R ji ( 2 z ) K j ij 2 2 � � � � z − 1 z + 1 K < ij > ( z ) = Q ( l ) R ji ( 2 z ) − 1 K j Q ( l ) K ( z ) = ij K i ij . 2 2 Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

  83. Physical motivations and framework Fusion procedure Integrable discrete-time dynamics Definition of the process Generalized exclusion processes “Fused” matrix ansatz A similar fusion procedure can also be applied to the K -matrices � � � � z − 1 z + 1 K < ij > ( z ) = Q ( l ) Q ( r ) K ( z ) = ij K i R ji ( 2 z ) K j ij 2 2 � � � � z − 1 z + 1 K < ij > ( z ) = Q ( l ) R ji ( 2 z ) − 1 K j Q ( l ) K ( z ) = ij K i ij . 2 2 4 cz � ( 2 z − 1 )( c − a )+ 2 �   8 c 2 z ( 2 z − 1 ) � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � ∗   8 az � ( 2 z − 1 )( c − a )+ 2 � 8 cz � ( 2 z − 1 )( a − c )+ 2 �       � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � K ( z ) = ∗     4 az � ( 2 z − 1 )( a − c )+ 2 �   8 a 2 z ( 2 z − 1 ) � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � � ( 2 z − 1 )( a + c )+ 2 �� ( 2 z + 1 )( a + c )+ 2 � ∗ Matthieu VANICAT Integrable discrete-time dynamics, generalized exclusion processes

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Generalized Gaussian wave packet dynamics: integrable and chaotic systems Steven Tomsovic

Generalized Gaussian wave packet dynamics: integrable and chaotic systems Steven Tomsovic

Wave packets Real/complex trajectories Kicked rotor Generalized Gaussian wave packet dynamics: integrable and chaotic systems Steven Tomsovic Washington State University, Pullman, WA USA work supported by: US National Science Foundation

640 views • 19 slides
Integrable gap probabilities for the Generalized Bessel process Manuela Girotti SISSA,

Integrable gap probabilities for the Generalized Bessel process Manuela Girotti SISSA,

Integrable gap probabilities for the Generalized Bessel process Integrable gap probabilities for the Generalized Bessel process Manuela Girotti SISSA, Trieste, June 7th 2017 1 / 27 Integrable gap probabilities for the

877 views • 35 slides
Invariant measures for KdV and Toda-type discrete integrable systems Online Open Probability

Invariant measures for KdV and Toda-type discrete integrable systems Online Open Probability

Invariant measures for KdV and Toda-type discrete integrable systems Online Open Probability School 12 June 2020 David Croydon (Kyoto) joint with Makiko Sasada (Tokyo) and Satoshi Tsujimoto (Kyoto) 1. KDV AND TODA-TYPE DISCRETE INTEGRABLE

471 views • 32 slides
Simple Exclusion Process conditioned on extreme flux V. Popkov Universit di Salerno, Italy

Simple Exclusion Process conditioned on extreme flux V. Popkov Universit di Salerno, Italy

Effective Dynamics of Asymmetric Simple Exclusion Process conditioned on extreme flux V. Popkov Universit di Salerno, Italy with G. Schtz , D. Simon MPI, Drezden, July 2011 Plan Introduction TASEP on a ring as an integrable model

456 views • 24 slides
Quench, Hydro and Floquet dynamics in integrable systems RAQIS, Annecy, 12 September 2018

Quench, Hydro and Floquet dynamics in integrable systems RAQIS, Annecy, 12 September 2018

Quench, Hydro and Floquet dynamics in integrable systems RAQIS, Annecy, 12 September 2018 Jean-Sbastien Caux Universiteit van Amsterdam Plan of the talk Equilibrium dynamics Out-of-equilibrium dynamics Quenches Quasisolitons and

1.31k views • 46 slides
Discrete line complexes and integrable evolution of minors by W.K. Schief The University of New

Discrete line complexes and integrable evolution of minors by W.K. Schief The University of New

Discrete line complexes and integrable evolution of minors by W.K. Schief The University of New South Wales, Sydney ARC Centre of Excellence for Mathematics and Statistics of Complex Systems [with A.I. Bobenko] 1. The algebraic set-up For the

372 views • 20 slides
12 Networked Control Systems: a Discrete-time Approach Nathan van de Wouw Dynamics and Control

12 Networked Control Systems: a Discrete-time Approach Nathan van de Wouw Dynamics and Control

12 Networked Control Systems: a Discrete-time Approach Nathan van de Wouw Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands Universit Catholique de Louvain, October 5th,

1.42k views • 47 slides
Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Systems and

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Systems and

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Stochastic Systems Reading: Section 1.4 in Shedler or Section 4.1 in Haas The GSMP Model Simulating GSMPs

313 views • 7 slides
Discrete-time Systems in the Time Domain Chaiwoot Boonyasiriwat August 21, 2020 Discrete-time

Discrete-time Systems in the Time Domain Chaiwoot Boonyasiriwat August 21, 2020 Discrete-time

Discrete-time Systems in the Time Domain Chaiwoot Boonyasiriwat August 21, 2020 Discrete-time Systems A discrete-time system is an entity that processes a discrete-time input signal to produce a discrete- time output signal

851 views • 62 slides
A master solution of the Yang-Baxter equation and classical discrete integrable equations.

A master solution of the Yang-Baxter equation and classical discrete integrable equations.

A master solution of the Yang-Baxter equation and classical discrete integrable equations. Vladimir Bazhanov (in collaboration with Sergey Sergeev &amp; Vladimir Mangazeev) Australian National University Mathematical Statistical Mechanics,

808 views • 39 slides
Discrete-Time Processing Overview Introduction Review of key discrete-time concepts

Discrete-Time Processing Overview Introduction Review of key discrete-time concepts

Discrete-Time Processing Overview Introduction Review of key discrete-time concepts Definitions Few examples Basic mathematical signals Assume you are familiar with most of these ideas Signal types Discrete-time

375 views • 18 slides
CS 220: Discrete Structures and their Applications counting by complement, inclusion exclusion,

CS 220: Discrete Structures and their Applications counting by complement, inclusion exclusion,

CS 220: Discrete Structures and their Applications counting by complement, inclusion exclusion, the pigeonhole principle zybooks 7.8 7.10 example problem How many 6-bit strings have at least one 0? You can count them directly: Number of

424 views • 15 slides
Cyber-Physical Systems Discrete Dynamics ICEN 553/453 Fall 2018 Prof. Dola Saha 1 Discrete

Cyber-Physical Systems Discrete Dynamics ICEN 553/453 Fall 2018 Prof. Dola Saha 1 Discrete

Cyber-Physical Systems Discrete Dynamics ICEN 553/453 Fall 2018 Prof. Dola Saha 1 Discrete Systems Discrete = individually separate / distinct A discrete system is one that operates in a sequence of discrete steps or has signals

500 views • 27 slides
Cyber-Physical Systems Discrete Dynamics IECE 553/453 Fall 2019 Prof. Dola Saha 1 Discrete

Cyber-Physical Systems Discrete Dynamics IECE 553/453 Fall 2019 Prof. Dola Saha 1 Discrete

Cyber-Physical Systems Discrete Dynamics IECE 553/453 Fall 2019 Prof. Dola Saha 1 Discrete Systems Discrete = individually separate / distinct A discrete system is one that operates in a sequence of discrete steps or has signals

825 views • 29 slides
Discrete analytic functions. Integrable structure Alexander Bobenko Technical University Berlin

Discrete analytic functions. Integrable structure Alexander Bobenko Technical University Berlin

Discrete analytic functions. Integrable structure Alexander Bobenko Technical University Berlin Geometry Conference in honour of Nigel Hitchin, Madrid, September 4-8, 2006 DFG Research Unit 565 Polyhedral Surfaces Alexander Bobenko

851 views • 39 slides
Generalized Solutions of Riccati equations and inequalities D.Z. Arov, M.A. Kaashoek, D.R. Pik

Generalized Solutions of Riccati equations and inequalities D.Z. Arov, M.A. Kaashoek, D.R. Pik

Generalized Solutions of Riccati equations and inequalities D.Z. Arov, M.A. Kaashoek, D.R. Pik August 2017 1 /30 Time-invariant system Time-invariant system with discrete time n A, B, C, D are bounded linear operators between Hilbert spaces.

452 views • 30 slides
1 R O M I T C H A K R A B O R T Y T H E M A Z Z I O T T I G R O U P T H E U N I V E R S I T

1 R O M I T C H A K R A B O R T Y T H E M A Z Z I O T T I G R O U P T H E U N I V E R S I T

OPENNESS OF MANY-ELECTRON QUANTUM SYSTEMS FROM THE GENERALIZED PAULI EXCLUSION PRINCIPLE 1 R O M I T C H A K R A B O R T Y T H E M A Z Z I O T T I G R O U P T H E U N I V E R S I T Y O F C H I C A G O THE PAULI EXCLUSION PRINCIPLE No

495 views • 36 slides
Scott Ranks of Models of a Theory Matthew Harrison-Trainor University of California, Berkeley

Scott Ranks of Models of a Theory Matthew Harrison-Trainor University of California, Berkeley

Scott Ranks of Models of a Theory Matthew Harrison-Trainor University of California, Berkeley Notre Dame, September 2015 Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 1 / 38 Overview The Scott rank of a

1.04k views • 85 slides
Probability and Random Processes Lecture 11 Measurable dynamical systems Random processes

Probability and Random Processes Lecture 11 Measurable dynamical systems Random processes

Probability and Random Processes Lecture 11 Measurable dynamical systems Random processes as dynamical systems Stationarity Ergodic theory Mikael Skoglund, Probability and random processes 1/13 Measurable Dynamical System A

250 views • 7 slides
Motion in more than one dimension Vector equations of motion Different components (dimensions)

Motion in more than one dimension Vector equations of motion Different components (dimensions)

Motion in more than one dimension Vector equations of motion Different components (dimensions) coupled through F Example: Planetary motion (in a 2D plane) Gravitational force: Two-body problem; can be reduced to one-body problem for effective

681 views • 14 slides
Dynamics of open quantum systems via resonances Marco Merkli Deptartment of Mathematics and

Dynamics of open quantum systems via resonances Marco Merkli Deptartment of Mathematics and

Dynamics of open quantum systems via resonances Marco Merkli Deptartment of Mathematics and Statistics Memorial University CQIQC-V, August 14, 2013 Open quantum systems System + Environment models Hamiltonian H = H S + H R + V H S =

319 views • 15 slides
Solutions for a hyperbolic model of multiphase flow and related numerical issues Debora Amadori

Solutions for a hyperbolic model of multiphase flow and related numerical issues Debora Amadori

Solutions for a hyperbolic model of multiphase flow and related numerical issues Debora Amadori University of LAquila (Italy) AMIS2012, June 20 2012 Outline Outline Part I: A multiphase flow model 1 Description Basics on the homogeneous

745 views • 62 slides
Sound Multiple Choice Practice Problems Problems Slide 3 / 51 Slide 4 / 51 1 Two sound

Sound Multiple Choice Practice Problems Problems Slide 3 / 51 Slide 4 / 51 1 Two sound

Slide 1 / 51 Slide 2 / 51 Sound Multiple Choice Practice Problems Problems Slide 3 / 51 Slide 4 / 51 1 Two sound sources S 1 and S 2 produce waves with 2 Which of the following is a true statement about frequencies 500 Hz and 250 Hz. When

695 views • 9 slides
Pasadena, USA, December 9 14, 2001 RFNC-VNIITF MULTIFUNCTIONAL SHOCK TUBE FOR INVESTIGATING

Pasadena, USA, December 9 14, 2001 RFNC-VNIITF MULTIFUNCTIONAL SHOCK TUBE FOR INVESTIGATING

8 th International Workshop on Physics of Compressible Turbulent Mixing Pasadena, USA, December 9 14, 2001 RFNC-VNIITF MULTIFUNCTIONAL SHOCK TUBE FOR INVESTIGATING THE EVOLUTION OF INSTABILITIES IN UNSTATIONARY GASDYNAMIC FLOWS Yu. A.

314 views • 7 slides

More recommend