fluctuations and skewness of the current in the partially asymmetric exclusion process we use functional bethe ansatz equations to calculate the cumulants of the total current in the partially asymmetric exclusion process we recover known formulae for the first two cumulants mean value of the current and diffusion constant and obtain an explicit finite size formula for the third cumulant the expression for the third cumulant takes a simple integral form in the limit where the asymmetry scales as the inverse of the square root of the size of the system which corresponds to a natural separation between weak and strong asymmetry

exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

transfer matrices for the totally asymmetric simple exclusion process we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries we show using the recursive structure of the markov matrix that encodes the dynamics that there exist two transfer matrices tl 1 l and that intertwine the markov matrices of consecutive system sizes this semi conjugation property of the dynamics provides an algebraic counterpart for the matrix product representation of the steady state of the process

spectral gap of the totally asymmetric exclusion process at arbitrary filling we calculate the spectral gap of the markov matrix of the totally asymmetric simple exclusion process tasep on a ring of l sites with n particles our derivation is simple and self contained and extends a previous calculation that was valid only for half filling we use a special property of the bethe equations for tasep to reformulate them as a one body problem our method is closely related to the one used to derive exact large deviation functions of the tasep

stationary distributions of multi type totally asymmetric exclusion processes we consider totally asymmetric simple exclusion processes with n types of particle and holes n taseps on z and on the cycle z n angel recently gave an elegant construction of the stationary measures for the 2 tasep based on a pair of independent product measures we show that angel s construction can be interpreted in terms of the operation of a discrete time m m 1 queueing server the two product measures correspond to the arrival and service processes of the queue we extend this construction to represent the stationary measures of an n tasep in terms of a system of queues in tandem the proof of stationarity involves a system of n 1 taseps whose evolutions are coupled but whose distributions at any fixed time are independent using the queueing representation we give quantitative results for stationary probabilities of states of the n tasep on z n and simple proofs of various independence and regeneration properties for systems on z

exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra we study the partially asymmetric exclusion process with open boundaries we generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q due to the relationship between the matrix algebra and the q deformed quantum harmonic oscillator algebra we find that q hermite polynomials along with their orthogonality properties and generating functions are of great utility we employ two distinct sets of q hermite polynomials one for q 1 it turns out that these correspond to two distinct regimes the previously studied case of forward bias q 1 where the boundaries support a current opposite in direction to the bulk bias for the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size

matrix product representation of the stationary state of the open zero range process many one dimensional lattice particle models with open boundaries like the paradigmatic asymmetric simple exclusion process asep have their stationary states represented in the form of a matrix product with matrices that do not explicitly depend on the lattice site in contrast the stationary state of the open 1d zero range process zrp takes an inhomogeneous factorized form with site dependent probability weights we show that in spite of the absence of correlations the stationary state of the open zrp can also be represented in a matrix product form where the matrices are site independent non commuting and determined from algebraic relations resulting from the master equation we recover the known distribution of the open zrp in two different ways first using an explicit representation of the matrices and boundary vectors second from the sole knowledge of the algebraic relations satisfied by these matrices and vectors finally an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden markov chains

multiline queues with spectral parameters using the description of multiline queues as functions on words we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues we show that the spectral weight of a word is invariant under a natural action of the symmetric group giving a proof of the commutativity conjecture of arita ayyer mallick and prolhac we give a determinant formula for the spectral weight of a word which gives a proof of a conjecture of the first author and linusson

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

bethe ansatz and q operator for the open asep in this paper we look at the asymmetric simple exclusion process with open boundaries with a current counting deformation we construct a two parameter family of transfer matrices which commute with the deformed markov matrix of the system we show that these transfer matrices can be factorized into two commuting matrices with one parameter each which can be identified with baxter s q operator and that for certain values of the product of those parameters they decompose into a sum of two commuting matrices one of which is the usual one parameter transfer matrix for a given dimension of the auxiliary space using this we find the t q equation for the open asep and through functional bethe ansatz techniques we obtain an exact expression for the dominant eigenvalue of the deformed markov matrix

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

multispecies tasep and the tetrahedron equation we introduce a family of layer to layer transfer matrices in a three dimensional 3d lattice model which can be viewed as partition functions of the q oscillator valued six vertex model on an m n square lattice by invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions at q 0 and m n they ultimately yield a new proof of the steady state formula for the n species totally asymmetric simple exclusion process obtained recently by the authors revealing the 3d integrability in the matrix product construction

von neumann algebra invariants of dirac operators abstract in this paper we define and study certain von neumann algebra invariants associated to the dirac operator acting on l 2 spinors on the universal covering space of a compact riemannian spin manifold we first study a novikov shubin type invariant which is a conformal invariant but which is not independent of the choice of metric however we prove results which give evidence that this invariant may always be positive when the novikov shubin type invariant is positive we can define the von neumann algebra determinant of the dirac laplacian and the corresponding element in the determinant line of the space of l 2 harmonic spinors on the universal covering space which we also compute for certain locally symmetric spaces finally we study a von neumann algebra eta invariant associated to the dirac operator and we show that it is sometimes an obstruction to the existence of metrics of positive scalar curvature

the spinor heat kernel in maximally symmetric spaces the heat kernelk x x t of the iterated dirac operator on ann dimensional simply connected maximally symmetric riemannian manifold is calculated on the odd dimesional hyperbolic spacesk is a minakshisundaram dewitt expansion which terminates to the coefficientan 1 2 and is exact on the odd spheres the heat kernel may be written as an image sum of wkb kernels each term corresponding to a classical path geodesic in the even dimensional case the wkb approximation is not exact but a closed form ofk is derived both in terms of spherical eigenfunctions and of a sum over classical paths the spinor plancherel measure and function in the hyperbolic case are also calculated a simple relation between the analytic structure of onhn and the degeneracies of the dirac operator onsn is found

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

matrix product solution of an inhomogeneous multi species tasep we study a multi species exclusion process with inhomogeneous hopping rates and find a matrix product representation for the stationary state of this model the matrices belong to the tensor algebra of the fundamental quadratic algebra associated with the exclusion process we show that our matrix product representation is equivalent to a graphical construction proposed by ayyer and linusson 2012 arxiv 1206 0316 which generalizes an earlier probabilistic construction due to ferrari and martin 2007 ann prob 35 807

intertwining operators for solving differential equations with applications to symmetric spaces the use of intertwining operators to solve both ordinary and partial differential equations is developed classes of intertwining operators are constructed which transform between laplacians which are self adjoint with respect to different non trivial measures in the two dimensional case the intertwining operator transforms a non separable partial differential operator to a separable one as an application the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed

an exact formula for the statistics of the current in the tasep with open boundaries we study the totally asymmetric exclusion process on a finite one dimensional lattice with open boundaries i e in contact with two reservoirs at different potentials the total time integrated current through the system is a random variable that scales linearly with time in the long time limit we conjecture a parametric representation for the generating function of the cumulants of the current which is related to the large deviation function by laplace transform this formula is valid for all system sizes and for all values of the boundary coupling parameters

headway oscillations and phase transitions for diffusing particles with increased velocity a totally asymmetric exclusion process with n particles on a periodic one dimensional lattice of l sites is considered where particles can move one or two sites per infinitesimal timestep an exact analysis for n 2 and a mean field theory in comparison with simulations show even odd oscillations in the headway distribution of particles the expression headway is understood as the number of empty sites in front of a particle oscillations become maximal if particles only move at their maximum possible speed a phase transition separates two density profiles around a generated perturbation that plays the role of a defect the matrix product ansatz is generalized to obtain the exact solution for finite n and l thermodynamically the headway distribution yields the mean field result as n while it is not described generally by a product measure

unified trade off optimization for general heat devices with nonisothermal processes an analysis of the efficiency and coefficient of performance cop for general heat engines and refrigerators with nonisothermal processes is conducted under the trade off criterion the specific heat of the working medium has significant impacts on the optimal configurations of heat devices for cycles with constant specific heat the bounds of the efficiency and cop are found to be the same as those obtained through the endoreversible carnot ones however they are independent of the cycle time durations for cycles with nonconstant specific heat whose dimensionless contact time approaches infinity the general alternative upper and lower bounds of the efficiency and cop under the trade off criteria have been proposed under the asymmetric limits furthermore when the dimensionless contact time approaches zero the endoreversible carnot model is recovered in addition the efficiency and cop bounds of different kinds of actual heat engines and refrigerators have also been analyzed this paper may provide practical insight for designing and operating actual heat engines and refrigerators

exact diffusion constant for the one dimensional partially asymmetric exclusion model we calculate exactly the diffusion constant associated with the fluctuations of the current for the partial asymmetric exclusion model on a ring with an arbitrary number of particles and holes we also give the diffusion constant of a tagged particle on that ring our approach extends using the deformed harmonic oscillator algebra a result already known for the fully asymmetric case in the limit of weak asymmetry we extract from our exact expression the crossover between the edwards wilkinson and the kardar parisi zhang equations in 1 1 dimensions

matrix representation of the stationary measure for the multispecies tasep in this work we construct the stationary measure of the n species totally asymmetric simple exclusion process in a matrix product formulation we make the connection between the matrix product formulation and the queueing theory picture of ferrari and martin in particular in the standard representation the matrices act on the space of queue lengths for n 2 the matrices in fact become tensor products of elements of quadratic algebras this enables us to give a purely algebraic proof of the stationary measure which we present for n 3

the asymmetric simple exclusion process an integrable model for non equilibrium statistical mechanics the asymmetric simple exclusion process asep plays the role of a paradigm in non equilibrium statistical mechanics we review exact results for the asep obtained by the bethe ansatz and put emphasis on the algebraic properties of this model the bethe equations for the eigenvalues of the markov matrix of the asep are derived from the algebraic bethe ansatz using these equations we explain how to calculate the spectral gap of the model and how global spectral properties such as the existence of multiplets can be predicted an extension of the bethe ansatz leads to an analytic expression for the large deviation function of the current in the asep that satisfies the gallavotti cohen relation finally we describe some variants of the asep that are also solvable by the bethe ansatz

forms on vector bundles over hyperbolic manifolds and the trace anomaly we study gauge theories based on abelian p forms on real compact hyperbolic manifolds an explicit formula for the trace anomaly corresponding to skew symmetric tensor fields is obtained by using zeta function regularization and the trace tensor kernel formula explicit exact and numerical values of the anomaly for p forms of order up to p 4 in spaces of dimension up to n 10 are then calculated

exact solution to integrable open multi species ssep and macroscopic fluctuation theory we introduce a multi species generalization of the symmetric simple exclusion process with open boundaries this model possesses the property of being integrable and appears as physically relevant because the boundary conditions can be interpreted as the interaction with particles reservoirs with fixed densities of each species the system is driven out of equilibrium by these reservoirs the steady state is analytically computed in a matrix product form this algebraic structure allows us to obtain exact expressions for the mean particle currents and for the one and two point correlation functions an additivity principle is also derived from the matrix ansatz and permits the computation of the large deviation functional of the density profile we also propose a description of the model in the context of the macroscopic fluctuation theory and we check the consistency with the exact computations from the finite size lattice

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

exact solution of the totally asymmetric simple exclusion process shock profiles the microscopic structure of macroscopic shocks in the one dimensional totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles first and second class this nonequilibrium steady state factorizes about any second class particle which implies factorization in the one component system about the random shock position it also exhibits several other interesting features including long range correlations in the limit of zero density of the second class particles the solution also shows that a finite number of second class particles in a uniform background of first class particles form a weakly bound state

exact solution of an exclusion process with three classes of particles and vacancies we present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies using a matrix product ansatz we find exact expressions for the weights of the configurations in the stationary state the solution involves tensor products of quadratic algebras

exact diffusion constant for one dimensional asymmetric exclusion models the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions is considered on a ring of size n the steady state of this system is known all configurations have equal weight which allows for easy computation of the average velocity of a particle in the steady state here an exact expression for the diffusion constant of a particle is obtained for arbitrary number of particles and system size by using a matrix formulation two limits of infinite system size n are discussed firstly when the number of particles remains finite as n to infinity the diffusion constant remains dependent on the exact number of particles due to correlations between successive collisions secondly when the density p of particles is non zero i e when the number of particles is equal to n rho as n to infinity the diffusion constant scales as n 12 the exponent 1 2 is related to the dynamic exponent z 3 2 of the kpz equation in 1 1 dimensions

brownian bridges for late time asymptotics of kpz fluctuations in finite volume height fluctuations are studied in the one dimensional totally asymmetric simple exclusion process with periodic boundaries with a focus on how late time relaxation towards the non equilibrium steady state depends on the initial condition using a reformulation of the matrix product representation for the dominant eigenstate the statistics of the height at large scales is expressed for arbitrary initial conditions in terms of extremal values of independent standard brownian bridges comparison with earlier exact bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non intersecting brownian bridges with exponentially distributed distances between the endpoints

matrix product formula for u _ q a_ 2 1 zero range process the zero range processes introduced recently by mangazeev maruyama and the authors are integrable discrete and continuous time markov processes associated with the stochastic r matrix derived from the well known quantum r matrix by constructing a representation of the relevant zamolodchikov faddeev algebra we present for n 2 a matrix product formula for the steady state probabilities in terms of q boson operators

current fluctuations in the exclusion process and bethe ansatz we use the bethe ansatz to derive analytical expressions for the current statistics in the asymmetric exclusion process with both forward and backward jumps the bethe equations are highly coupled and this fact has impeded their use in deriving exact results for finite systems we overcome this technical difficulty by reformulating bethe equations into a one variable polynomial problem akin to the functional bethe ansatz the perturbative solution of this equation leads to the cumulants of the current we calculate here the first two orders and derive exact formulae for the mean value of the current and its fluctuations

the matrix product solution of the multispecies partially asymmetric exclusion process we find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles the solution is in the form of a matrix product representation where the matrices for a system of n species are defined recursively in terms of the matrices for a system of n 1 species a complete proof is given based on the quadratic relations verified by these matrices this matrix product construction is interpreted in terms of the action of a transfer matrix

remarks on the multi species exclusion process with reflective boundaries we investigate one of the simplest multi species generalizations of the one dimensional exclusion process with reflective boundaries the markov matrix governing the dynamics of the system splits into blocks sectors specified by the number of particles of each kind we find matrices connecting the blocks in a matrix product form the procedure generalized matrix ansatz to verify that a matrix intertwines blocks of the markov matrix was introduced in the periodic boundary condition arita et al 2011 j phys a 44 335004 which starts with a local relation the solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

integrable boundary conditions for multi species asep the first result of the present paper is to provide classes of explicitsolutions for integrable boundary matrices for the multi species asep with anarbitrary number of species all the solutions we have obtained can be seen as representations of a newalgebra that contains the boundary hecke algebra the boundary hecke algebra isnot sufficient to build these solutions this is the second result of ourpaper

finite dimensional representations of the quadratic algebra applications to the exclusion process we study the one dimensional partially asymmetric simple exclusion process asep with open boundaries that describes a system of hard core particles hopping stochastically on a chain coupled to reservoirs at both ends derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra closely related to the deformed oscillator algebra we construct all finite dimensional irreducible representations of this algebra this enables us to compute the stationary bulk density as well as all correlation lengths for the asep on a set of special curves of the phase diagram

recursive structures in the multispecies tasep we consider a multispecies generalization of the totally asymmetric simple exclusion process tasep with the simple hopping rule for the and th class particles the transition occurs with a rate independent from the values and ferrari and martin 2007 ann prob 35 807 obtained the stationary state of this model thanks to a combinatorial algorithm which was subsequently interpreted as a matrix product representation by evans et al 2009 j stat phys 135 217 this matrix ansatz shows that the stationary state of the multispecies tasep with n classes of particles n tasep can be constructed algebraically by the action of an operator on the n 1 tasep stationary state besides arita et al 2009 j phys a math theor 45 345002 analyzed the spectral structure of the markov matrix they showed that the set of eigenvalues of the n tasep contains those of the n 1 tasep and that the various spectral inclusions can be encoded in a hierarchical set theoretic structure known as the hasse diagram inspired by these works we define nontrivial operators that allow us to construct eigenvectors of the n tasep by lifting the eigenvectors of the n 1 tasep this goal is achieved by generalizing the matrix product representation and the ferrari martin algorithm in particular we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines markov matrices of different values of n

bethe ansatz solution for a defect particle in the asymmetric exclusion process the asymmetric exclusion process on a ring in one dimension is considered with a single defect particle the steady state has previously been solved by a matrix product method here we use the bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle this allows us to recover steady state properties known from the matrix approach such as the velocity and obtain new results such as the diffusion constant of the defect particle in the case where the defect particle is a second class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is gaussian moreover the variance diffusion constant grows as l1 2 where l is the system size this behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system in the case where the defect particle produces a shock our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by ferrari and fontes

eta invariant and selberg zeta function of odd type over convex co compact hyperbolic manifolds abstract we show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type z o associated to the spinor bundle on an odd dimensional convex co compact hyperbolic manifold h 2 n 1 as a byproduct we do a full analysis of the spectral and scattering theory of the dirac operator on asymptotically hyperbolic manifolds we show that there is a natural eta invariant d associated to the dirac operator d over a convex co compact hyperbolic manifold h 2 n 1 and that exp i d z o 0 thus extending millson s formula to this setting under some assumption on the exponent of convergence of poincare series for the group we also define an eta invariant for the odd signature operator and we show that for schottky 3 dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the zograf factorization formula relating two natural kahler potentials for weil petersson metric on schottky space

matrix product formula for macdonald polynomials 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

general limit distributions for sums of random variables with a matrix product representation the general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized using a mapping to a hidden markov chain formalism non standard limit distributions are obtained and related to a form of ergodicity breaking in the underlying non homogeneous hidden markov chain the link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions

open two species exclusion processes with integrable boundaries we give a complete classification of integrable markovian boundary conditions for the asymmetric simple exclusion process with two species or classes of particles some of these boundary conditions lead to non vanishing particle currents for each species we explain how the stationary state of all these models can be expressed in a matrix product form starting from two key components the zamolodchikov faddeev and ghoshal zamolodchikov relations this statement is illustrated by studying in detail a specific example for which the matrix ansatz involving 9 generators is explicitly constructed and physical observables such as currents densities calculated

exact results for an asymmetric annihilation process with open boundaries we consider a nonequilibrium reaction diffusion model on a finite one dimensional lattice with bulk and boundary dynamics inspired by the glauber dynamics of the ising model we show that the model has a rich algebraic structure that we use to calculate its properties in particular we show that the markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes this remark leads us to devise a technique which we call the transfer matrix ansatz that allows us to determine the steady state distribution and correlation functions furthermore we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of markov matrices finally we compare the transfer matrix ansatz used here with the matrix product representation of the steady state of one dimensional stochastic models

on matrix product ground states for reaction diffusion models we discuss a new mechanism leading to a matrix product form for the stationary state of one dimensional stochastic models the corresponding algebra is quadratic and involves four different matrices for the example of a coagulation decoagulation model explicit four dimensional representations are given and exact expressions for various physical quantities are recovered we also find the general structure of n point correlation functions at the phase transition

generalized matrix ansatz in the multispecies exclusion process the partially asymmetric case we investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring this process has a rich combinatorial spectral structure and a matrix product form for the stationary state in the totally asymmetric case operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by arita et al 2011 j phys a math theor 44 335004 the existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra generalized matrix ansatz in this work we construct the family of representations explicitly for the partially asymmetric case this solution cannot be obtained by a simple deformation of the totally asymmetric case