We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. For totally asymmetric diffusion we calculate the spectral gap, which characterizes the approach to stationarity at large times. We observe boundary induced crossovers in and between massive, diffusive and KPZ scaling regimes. The partially asymmetric simple exclusion process (PASEP) describes the asymmetric diffusion of particles along a one-dimensional chain with L sites. It is one of the most studied models of non-equilibrium statistical mechanics, see [1,2] for recent reviews. This is in part due to the fact that is one of the simplest lattice gas models, but also because of its applicability to molecular diffusion in zeolites [3], bioploymers [4], traffic flow [5] and other one-dimensional complex systems [6].At large times the PASEP exhibits a relaxation towards a nonequilibrium stationary state. An interesting feature of the PASEP is the presence of boundary induced phase transitions [7]. In particular, in an open system with two boundaries at which particles are injected and extracted with given rates, the bulk behaviour in the stationary state is strongly dependent on the injection and extraction rates. Over the last decade many stationary state properties of the PASEP with open boundaries have been determined exactly [1,2,[8][9][10][11]. On the other hand, much less is known about its dynamics. This is in contrast to the PASEP on a ring for which exact results using Bethe's ansatz have been available for a long time [12,13]. For open boundaries there have been several studies of dynamical properties based mainly on numerical and phenomenological methods [14,15]. In this Letter we employ Bethe's ansatz to obtain exact results for the approach to stationarity at large times in the PASEP with open boundaries. Upon varying the boundary rates, we find crossovers in massive regions, with dynamic exponents z = 0, and between massive and scaling regions with diffusive (z = 2) and KPZ (z = 3/2) behaviour.The dynamical rules of the PASEP are as follows. At any given time t each site is either occupied by a particle or empty and the system evolves subject to the following rules. In the bulk of the system (i = 2, . . . , L − 1) a particle attempts to hop one site to the right with rate p and one site to the left with rate q. The hop is executed unless the neighbouring site is occupied, in which case nothing happens. On the first and last sites these rules are modified. If site i = 1 is empty, a particle may enter the system with rate α. If on the other hand site 1 is occupied by a particle, the latter will leave the system with rate γ. Similarly, at i = L particles are injected and extracted with rates δ and β respectively. With every site i we associate a Boolean variable τ i , indicating whether a particle is present (τ i = 1) or not (τ i = 0). The state of the system at time t is then characterized by the probability distribu...

We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at ∆ = − 1 2 , along with the groundstate wavefunction of the corresponding O(n) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes.

We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. By analysing these equations in detail for the cases of totally asymmetric and symmetric diffusion, we calculate the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at large times. In the totally asymmetric case we observe boundary induced crossovers between massive, diffusive and KPZ scaling regimes. We further study higher excitations, and demonstrate the absence of oscillatory behaviour at large times on the "coexistence line", which separates the massive low and high density phases. In the maximum current phase, oscillations are present on the KPZ scale t ∝ L −3/2 . While independent of the boundary parameters, the spectral gap as well as the oscillation frequency in the maximum current phase have different values compared to the totally asymmetric exclusion process with periodic boundary conditions. We discuss a possible interpretation of our results in terms of an effective domain wall theory. Exact Spectral Gaps of the Asymmetric Exclusion Process with Open Boundaries 2The partially asymmetric simple exclusion process (PASEP) [1, 2] is a model describing the asymmetric diffusion of hard-core particles along a one-dimensional chain with L sites. Over the last decade it has become one of the most studied models of non-equilibrium statistical mechanics, see [3,4] for recent reviews. This is due to its close relationship to growth phenomena and the KPZ equation [5], its use as a microscopic model for driven diffusive systems [6] and shock formation [7], its applicability to molecular diffusion in zeolites [8], biopolymers [9-11], traffic flow [12] and other one-dimensional complex systems [13].At large times the PASEP exhibits a relaxation towards a non-equilibrium stationary state. An interesting feature of the PASEP is the presence of boundary induced phase transitions [14]. In particular, in an open system with two boundaries at which particles are injected and extracted with given rates, the bulk behaviour in the stationary state is strongly dependent on the injection and extraction rates. Over the last decade many stationary state properties of the PASEP with open boundaries have been determined exactly [3,4,15,16,[19][20][21][22].On the other hand, much less is known about its dynamics. This is in contrast to the PASEP on a ring for which exact results using Bethe's ansatz have been available for a long time [23][24][25]. For open boundaries there have been several studies of dynamical properties based mainly on numerical and phenomenological methods [26][27][28][29][30]. Very recently a real-space renormalization group approach was introduced, which allows for the determination of the dynamical exponents [31].In this work, elaborating on [32], we employ Bethe's ansatz to obtain exact results for the approach to stationarity at large times in the PASEP with open boundaries. Upo...

We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms.This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley-Lieb algebra. For a system of size L these representations are all of dimension 2 L and, for generic points of the algebra, equivalent. However at exceptional points they can possess different indecomposable structures.We study the centralizer of the one-boundary Temperley-Lieb algebra in the 'non-diagonal' spin-1 2 representation and find its eigenvalues and eigenvectors. In the exceptional cases the centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.

Conjectures for analytical expressions for correlations in the dense O(1) loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries "open". We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O(1) loop model and the XXZ spin chain for the different types of boundary conditions.

The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.

We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of the centre. For generic values of this parameter there is a unique representation of dimension 2 N and we show that it is isomorphic to a tensor space representation. We construct a basis in which the Gram matrix is diagonal and use this to discuss the irreducibility of this representation.

Abstract. We derive a matrix product formula for symmetric Macdonald polynomials.Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.