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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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