integrable xyz spin chain with boundaries we consider a general class of boundary terms of the open xyz spin 1 2 chain compatible with integrability we have obtained the general elliptic solution of a k matrix obeying the boundary yang baxter equation using the r matrix of the eight vertex model and derived the associated integrable spin chain hamiltonian

origin of the singular bethe ansatz solutions for the heisenberg xxz spin chain we investigate symmetry properties of the bethe ansatz wave functions for the heisenberg xxz spin chain the xxz hamiltonian commutes simultaneously with the shift operator t and the lattice inversion operator v in the space of 1 with the eigenvalue of t we show that the bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of t and v with quantum number 1 1 where is the eigenvalue of v therefore the bethe ansatz wave functions should be singular for nondegenerate eigenstates of the hamiltonian with quantum number 1 1 it is also shown that such states exist in any nontrivial down spin number sector and that the number of them diverges exponentially with the chain length c 2000 published by elsevier science b v all rights reserved

the conical point in the ferroelectric six vertex model we examine the last unexplored regime of the asymmetric six vertex model the low temperature phase of the so called ferroelectric model the original publication of the exact solution by sutherland yang and yang and various derivations and reviews published afterward do not contain many details about this regime we study the exact solution for this model by numerical and analytical methods in particular we examine the behavior of the model in the vicinity of an unusual coexistence point that we call the conical point this point corresponds to additional singularities in the free energy that were not discussed in the original solution we show analytically that at this point many polarizations coexist and that unusual scaling properties hold in its vicinity

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

multispecies extension of the solvable partially asymmetric reaction diffusion processes by considering the master equation of the partially asymmetric diffusion process on a one dimensional lattice the most general boundary condition i e interactions for the multispecies reaction diffusion processes is considered the resulting system has various interactions including diffusion to the left and right two particle interactions a a a a and the extended n particle drop push interactions to the left and right we obtain three distinct new models the conditions on reaction rates to ensure the solvability of the resulting models are obtained the two particle conditional probabilities are calculated exactly

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

finite size effects and infrared asymptotics of the correlation functions in two dimensions the finite size corrections approach to the calculation of the asymptotics of correlation functions containing the oscillating terms in the asymptotics in 1 1 dimensional theories is presented explicit formulae for the critical exponents in integrable models are obtained

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

k theoretic boson fermion correspondence and melting crystals we study non hermitian integrable fermion and boson systems from the perspectives of grothendieck polynomials the models considered in this article are the five vertex model as a fermion system and the non hermitian phase model as a boson system both models are characterized by different solutions satisfying the same yang baxter relation from our previous works on the identification between the wavefunctions of the five vertex model and grothendieck polynomials we introduce skew grothendieck polynomials and derive the addition theorem among them using these relations we derive the wavefunctions of the non hermitian phase model as a determinant form which can also be expressed as grothendieck polynomials namely we establish a k theoretic boson fermion correspondence at the level of wavefunctions as a by product the partition function of the statistical mechanical model of a three dimensional 3d melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model the resultant expression can be regarded as a k theoretic generalization of the macmahon function describing the generating function of the plane partitions which interpolates the generating functions of two dimensional 2d and 3d young diagrams

universal properties in the free energy of the asymmetric n state vertex model electric field dependent free energy of the exactly solvable asymmetric n state vertex model i e in an arbitrary vertical and an arbitrary horizontal electric field v and h in the low temperature antiferroelectric phase is obtained in the form of the expansion in terms of small but nonzero polarization p exact mapping onto a microscopic surface model solid on solid sos model is employed to study the vicinal surface free energy below the roughening temperature tr which determines the equilibrium crystal shape ecs near the facet edge of a crystal the obtained expansion of the free energy is of the well established gruber mullins pokrovsky talapov gmpt type f p h f 0 h a h p b h p3 o p4 it is found that the coefficients a h and b h are identical with those of the asymmetric six vertex model based on this expansion in cooperation with the andreev construction of the ecs universal properties are verified along the whole facet contour first directly from the gmpt type expansion the critical exponents governing the rounding of the crystal facet are obtained the exponent is 3 2 in all direction except the tangential one in which case it is 3 second the universal relation between a h and b h is verified leading to the universal gaussian curvature jump at the facet edge and a universal relation between the critical amplitudes of the ecs profiles of the vicinal surface

from duality to determinants for q tasep and asep we prove duality relations for two interacting particle systems the q deformed totally asymmetric simple exclusion process q tasep and the asymmetric simple exclusion process asep expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents respectively duality implies that they solve systems of odes these systems are integrable and for particular step and half stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems solutions and hence also the moments we form laplace transform like generating functions of these moments and via residue calculus we compute two different types of fredholm determinant formulas for such generating functions for asep the first type of formula is new and readily lends itself to asymptotic analysis as necessary to reprove gue tracy widom distribution fluctuations for asep while the second type of formula is recognizable as closely related to tracy and widom s asep formula comm math phys 279 2008 815 844 j stat phys 132 2008 291 300 comm math phys 290 2009 129 154 j stat phys 140 2010 619 634 for q tasep both formulas coincide with those computed via borodin and corwin s macdonald processes probab theory related fields 2014 158 225 400 both q tasep and asep have limit transitions to the free energy of the continuum directed polymer the logarithm of the solution of the stochastic heat equation or the hopf cole solution to the kardar parisi zhang equation thus q tasep and asep are integrable discretizations of these continuum objects the systems of odes associated to their dualities are deformed discrete quantum delta bose gases and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics

relaxation times in the asep model using a dmrg method we compute the largest relaxation times for the totally asymmetric exclusion process tasep with open boundary conditions with a dmrg method this allows us to reach much larger system sizes than in previous numerical studies we are then able to show that the phenomenological theory of the domain wall indeed predicts correctly the largest relaxation time for large systems besides we can obtain results even when the domain wall approach breaks down and show that the kpz dynamical exponent z 3 2 is recovered in the whole maximal current phase

roughness of two nonintersecting one dimensional interfaces the dynamics of two spatially discrete one dimensional single step model interfaces with a noncrossing constraint is studied in both nonsymmetric propagating and symmetric relaxing cases we consider possible scaling scenarios and study a few special cases by using continuous time monte carlo simulations the roughness of the interfaces is observed to be nonmonotonic as a function of time and in the stationary state it is nonmonotonic also as a function of the strength of the effective force driving the interfaces against each other this is related on the one hand to the reduction of the available configuration space and on the other hand to the ability of the interfaces to conform to each other

boundary k matrices for the xyz xxz and xxx spin chains the general solutions for the factorization equations of the reflection matrices k or theta for the eight vertex and six vertex models xyz xxz and xxx chains are found the associated integrable magnetic hamiltonians are derived explicitly finding families depending on several continuous as well as discrete parameters

deformed strings in the heisenberg model we investigate solutions to the bethe equations for the isotropic s 1 2 heisenberg chain involving complex string like rapidity configurations of arbitrary length going beyond the traditional string hypothesis of undeformed strings we describe a general procedure to construct eigenstates including strings with generic deformations discuss general features of these solutions and provide a number of explicit examples including complete solutions for all wavefunctions of short chains we finally investigate some singular cases and show from simple symmetry arguments that their contribution to zero temperature correlation functions vanishes

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

phase transitions in an exactly soluble one dimensional exclusion process we consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one dimensional chain of sites the particles are allowed to hop onto unoccupied sites to the right only for the special case of 1 the model was solved previously by derridaet al here we extend the solution to general the phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation

exact integrable spin chains and transfer matrices related to models with stochastic dynamics abstract we present a general family of one dimensional quantum hamiltonians and two dimensional transfer matrices that are exact integrable through the bethe ansatz and describe the time fluctuations of classical one dimensional stochastic models these quantum chains are higher spin generalizations of the standard xxz chain and the related stochastic models describe the asymmetric diffusion of particles where the particles have arbitrary sizes and belong to distinct classes with hierarchical order the transfer matrices we present are connected with new 10 vertex models that generalizes the standard six vertex model these transfer matrices describe the dynamics of of some stochastic cellular automata

frequentist performance of bayesian confidence intervals for comparing proportions in 2 2 contingency tables this article investigates the performance in a frequentist sense of bayesian confidence intervals cis for the difference of proportions relative risk and odds ratio in 2 x 2 contingency tables we consider beta priors logit normal priors and related correlated priors for the two binomial parameters the goal was to analyze whether certain settings for prior parameters tend to provide good coverage performance regardless of the true association parameter values for the relative risk and odds ratio we recommend tail intervals over highest posterior density hpd intervals for invariance reasons to protect against potentially very poor coverage probabilities when the effect is large it is best to use a diffuse prior and we recommend the jeffreys prior otherwise with relatively small samples bayesian cis using more informative even uniform priors tend to have poorer performance than the frequentist cis based on inverting score tests which perform uniformly quite well for these parameters

finite time current probabilities in the asymmetric exclusion process on a ring we calculate the time dependent probability distribution of current through a selected bond in the totally asymmetric exclusion process with periodic boundary conditions we derive a general formula for the probability that the integrated current exceeds a given value n at the moment of time t the formula is written in a form of a contour integral of a determinant of a toeplitz matrix transforming the determinant expression we obtain a generalization of the known formula derived by johansson for the infinite one dimensional lattice to check the general formula we consider the specific case corresponding to the probability of a minimal non zero current for this case we get an explicit analytical expression and analyze its asymptotics

operator content of the xxz chain the finite size scaling spectra of the xxz heisenberg chain are presented for even and odd numbers of sites respectively the operator content is given for free as well as for toroidal boundary conditions

bayesian sample size determination for case control studies case control studies are among the most commonly used means of assessing association between exposure and outcome sample size determination and the optimal control to case ratio are vital to the design of such studies in this article we investigate bayesian sample size determination and the control to case ratio for case control studies when interval estimation is the goal of the eventual statistical analysis in certain cases we are able to derive approximate closed form sample size formulas we also describe two monte carlo methods each of which provides a unified approach to the sample size problem because they may be applied to a wide range of interval based criteria we compare the accuracy of the different methods we also extend our methods to include cross sectional designs and designs for gene environment interaction studies

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

annihilation diffusion processes an exactly solvable model a family of diffusion annihilation processes is introduced which is exactly solvable this family contains parameters that control the diffusion and annihilation rates the solution is based on the bethe ansatz and using special boundary conditions to represent the reaction the processes are investigated both on the lattice and on the continuum special cases of this family of processes are the simple exclusion process and the drop push model

stochastic models on a ring and quadratic algebras the three species diffusion problem the stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations we consider only exclusion processes and restrict the type of algebras such that one has recurrence relations for traces of words of different lengths this is possible only if the rates satisfy certain compatibility conditions these conditions are derived and explicit representations of the generators of the quadratic algebras are given

exact results for one dimensional totally asymmetric diffusion models several types of totally asymmetric diffusion models with and without exclusion are considered for some models conditional probabilities of finding n particles on lattice sites at time t with initial occupation at time t 0 are expressed in a determinant form on the other hand the q boson totally asymmetric diffusion model is introduced which interpolates the free boson model and the model with exclusion like interaction the effects of the interaction are compared for the case of two particle diffusion

nonexistence of codimension one anosov flows on compact manifolds with boundary a flow is anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition an anosov flow is codimension one if its contracting or expanding direction is one dimensional examples of codimension one anosov flows on compact boundaryless manifolds can be exhibited in any dimension 3 in this paper we prove that there are no codimension one anosov flows on compact manifolds with boundary the proof uses an extension to flows of some results in hirsch on invariant subsets of hyperbolic sets essays on topology and related topics memoires dedies a georges de rham 1970 pp 126 135 related to question 10 b in palis and pugh fifty problems in dynamical systems in j palis c c pugh eds dynamical systems warwick 1974 proc sympos appl topology and dynamical systems univ warwick coventry 1973 1974 presented to e c zeeman on his fiftieth birthday lecture notes in mathematics vol 468 springer berlin 1975 pp 345 353

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

bethe ansatz solution of discrete time stochastic processes with fully parallel update we present the bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics the model depends on two parameters p the probability of single particle hopping and q the deformation parameter which in the general case q 1 is responsible for long range interaction between particles the particular case q 0 corresponds to the nagel schreckenberg traffic model with v max 1 as a result we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles for the case q 0 the result is obtained for arbitrary size of the lattice and number of particles in the general case we study the model in the scaling limit and obtain the universal form specific for the kardar parisi zhang universality class we describe the phase transition occurring in the limit p 1 when q 0

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

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

diffusion of two repulsive particles in a one dimensional lattice the problem of the lattice diffusion of two particles coupled by a contact repulsive interaction is solved by finding analytical expressions of the two body probability characteristic function the interaction induces anomalous drift with a vanishing velocity the average coordinate of each particle growing at large times as t1 2 the leading term of the mean square dispersions displays normal diffusion with a diffusion constant made smaller by the interaction by the non trivial factor 1 1 the space continuous limit taken from the lattice calculations allows one to establish connection with the standard problem of diffusion of a single fictitious particle constrained by a totally reflecting wall comparison between lattice and continuous results display marked differences for transient regimes relevant with regards to high time resolution experiments and in addition show that due to slowly decreasing subdominant terms lattice effects persist even at very large times

directed nonabelian sandpile models on trees we define two general classes of nonabelian sandpile models on directed trees or arborescences as models of nonequilibrium statistical physics unlike usual applications of the well known abelian sandpile model these models have the property that sand grains can enter only through specified reservoirs

stationary state skewness in kpz type growth stationary states in kpz type growth have interesting short distance properties we find that typically they are skewed and lack particle hole symmetry for example hill tops are typically flatter than valley bottoms and all odd moments of the height distribution function are non zero stationary state skewness can be turned on and off in the 1 1 dimensional restricted solid on solid rsos model we construct the exact stationary state for its master equation in a four dimensional parameter space in this state steps are completely uncorrelated familiar models such as the kim kosterlitz model lie outside this space and their stationary states are skewed we demonstrate using finite size scaling that the skewness diverges with systems size but such that the skewness operator is irrelevant in 1 1 dimensions with an exponent and that the kpz fixed point lies at zero skewness

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

spectra of non hermitian quantum spin chains describing boundary induced phase transitions the spectrum of the non hermitian asymmetric xxz chain with additional non diagonal boundary terms is studied the lowest lying eigenvalues are determined numerically for the ferromagnetic chain the temporal correlation length shows the same properties as the spatial correlation length for the antiferromagnetic chain we find a conformal invariant spectrum where the partition function corresponds to the one of a coulomb gas with only magnetic charges this is in contrast to the case of diagonal boundary terms where one finds only electric charges similar results are obtained in a toy model that can be diagonalized analytically in terms of free fermions

completeness of bethe s states for the generalized xxz model we study the bethe ansatz equations for a generalized xxz model on a one dimensional lattice assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of bethe s states for a generalized xxz model we find an exact form for the inverse matrix related with vacancy numbers and compute its determinant this inverse matrix has a tridiagonal form generalizing the cartan matrix of type a

the asymmetric six vertex model the exact solution of the asymmetric six vertex model published nearly without derivation by sutherlandet al in 1967 is rederived in detail the transfer matrix method and the bethe ansatz solution for the free energy which can be calculated from an integral equation are discussed for some special cases zero or maximal polarization the integral equation can be solved exactly in addition an asymptotic analysis valid for small but nonzero polarization is carried out the analytical properties of the results and their relevance for the bcsos model are discussed

algebraic bethe ansatz for singular solutions the bethe equations for the isotropic periodic spin 1 2 heisenberg chain with n sites have solutions containing i 2 that are singular both the corresponding energy and the algebraic bethe ansatz vector are divergent such solutions must be carefully regularized we consider a regularization involving a parameter that can be determined using a generalization of the bethe equations these generalized bethe equations provide a practical way of determining which singular solutions correspond to eigenvectors of the model

optimal sequential designs of case control studies fixed case control studies separately collect a case sample and a control sample with the two sample sizes being fixed prior to studies and sometimes arbitrarily chosen this often results in loss of efficiency of case control designs in terms of cost saving or time saving of the studies we study sequential case control designs and in connection with treatment allocation and stochastic approximation derive a simple sampling rule that leads to optimal case control designs some important issues such as fixed width confidence intervals and sequential tests of hypotheses with possible early stopping to save time or costs which cannot be answered with fixed case control designs are shown to be naturally solved with the derived optimal sequential case control designs

completeness of solutions of bethe s equations we consider the bethe equations for the isotropic spin 1 2 heisenberg quantum spin chain with periodic boundary conditions we formulate a conjecture for the number of solutions with pairwise distinct roots of these equations in terms of numbers of so called singular or exceptional solutions using homotopy continuation methods we find all such solutions of the bethe equations for chains of length up to 14 the numbers of these solutions are in perfect agreement with the conjecture we also discuss an indirect method of finding solutions of the bethe equations by solving the baxter t q equation we briefly comment on implications for thermodynamical computations based on the string hypothesis

bayesian analysis for a single 2 x 2 table the simple comparison of two binomial populations is frequently of interest in epidemiology when the domains are large for small domains however there are no exact methods except fisher s exact test a basic problem therefore is to compare two populations by assessing the difference between the proportions of individuals who possess a characteristic in the first and second populations when there is prior information we take the proportions to have independent conjugate beta distributions with known parameters thereby facilitating a bayesian analysis we consider bayesian inference on functions of the proportions and the three most common scalar measures used in epidemiology and health services research namely relative risk odds ratio and attributable risk we develop the highest density regions both exact and approximate for relative risk odds ratio and attributable risk in addition we consider the bayes factor for testing whether the model with a common proportion holds rather than one with distinct proportions using data from the population based worcester heart attack study we apply our methodology to study gender differences in the therapeutic management of patients with acute myocardial infarction ami by selected demographic and clinical characteristics the bayes factor the approximate and exact intervals generally suggest that there are no substantial differences in the pharmacologic management of males and females hospitalized with ami

a deformation of affine hecke algebra and integrable stochastic particle system we introduce a deformation of the affine hecke algebra of type which describes the commutation relations of the divided difference operators found by lascoux and schutzenberger and the multiplication operators making use of its representation we construct an integrable stochastic particle system it is a generalization of the q boson system due to sasamoto and wadati we also construct eigenfunctions of its generator using the propagation operator as a result we get the same eigenfunctions for the boson process obtained by povolotsky

the q pushasep a new integrable model for traffic in 1 1 dimension we introduce a new interacting stochastic particle system q pushasep which interpolates between the q tasep of borodin and corwin probab theory relat fields 158 1 2 225 400 2014 see also borodin et al ann probab 42 6 2314 2382 2014 borodin and corwin int math res not 2 499 537 2015 o connell and pei electron j probab 18 95 1 25 2013 borodin et al comput math 2013 and the q pushtasep introduced recently borodin and petrov adv math 2013 in the q pushasep particles can jump to the left or to the right and there is a certain partially asymmetric pushing mechanism present this particle system has a nice interpretation as a model of traffic on a one lane highway using the quantum many body system approach we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals we also discuss relevant fredholm determinantal formulas for the distribution of the location of each particle and connections of the model with a certain two sided version of macdonald processes and with the semi discrete stochastic heat equation

the model with boundaries part i diagonalization of the finite chain this is the first of three papers dealing with the xx finite quantum chain with arbitrary not necessarily hermitian boundary terms this extends previous work where the periodic or diagonal boundary terms were considered in order to find the spectrum and wavefunctions an auxiliary quantum chain is examined which is quadratic in fermionic creation and annihilation operators and hence diagonalizable the secular equation is in general complicated but several cases were found when it can be solved analytically for these cases the ground state energies are given the appearance of boundary states is also discussed and in view of the applications considered in the next papers the one and two point functions are expressed in terms of pfaffians

shocks in the asymmetric exclusion process in this paper we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers the initial distribution is a product measure with asymptotic density at and at earlier results described the limiting behavior in all cases except for 0 1 2 1 here we treat the exceptional case which is more delicate it corresponds to the one in which a shock wave occurs in an associated partial differential equation in the cases treated earlier the limit was an extremal invariant measure by contrast in the present case the limit is a mixture of two invariant measures our theorem resolves a conjecture made by the third author in 1975 7 the convergence proof is based on coupling and symmetry considerations

spectral theory for interacting particle systems solvable by coordinate bethe ansatz we develop spectral theory for the q hahn stochastic particle system introduced recently by povolotsky that is we establish a plancherel type isomorphism result that implies completeness and biorthogonality statements for the bethe ansatz eigenfunctions of the system owing to a markov duality with the q hahn tasep a discrete time generalization of tasep with particles jump distribution being the orthogonality weight for the classical q hahn orthogonal polynomials we write down moment formulas that characterize the fixed time distribution of the q hahn tasep with general initial data the bethe ansatz eigenfunctions of the q hahn system degenerate into eigenfunctions of other not necessarily stochastic interacting particle systems solvable by the coordinate bethe ansatz this includes the asep the asymmetric six vertex model and the heisenberg xxz spin chain all models are on the infinite lattice in this way each of the latter systems possesses a spectral theory too in particular biorthogonality of the asep eigenfunctions which follows from the corresponding q hahn statement implies symmetrization identities of tracy and widom for asep with either step or step bernoulli initial configuration as corollaries another degeneration takes the q hahn system to the q boson particle system dual to q tasep studied in detail in our previous paper 2013 thus at the spectral theory level we unify two discrete space regularizations of the kardar parisi zhang equation stochastic heat equation namely q tasep and asep

on the integrability of zero range chipping models with factorized steady states the conditions of the integrability of general zero range chipping models with factorized steady states which were proposed in evans et al 2004 j phys a math gen 37 l275 are examined we find a three parametric family of hopping probabilities for the models solvable by the bethe ansatz which includes most of known integrable stochastic particle models as limiting cases the solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations which is proved as a byproduct we use the bethe ansatz to solve an eigenproblem for the transition matrix of the markov process on its basis we conjecture an integral formula for the green function of the evolution operator for the model on an infinite lattice and derive the bethe equations for the spectrum of the model on a ring

entanglement entropy of non unitary conformal field theory here we show that the renyi entanglement entropy of a region of large size l in a one dimensional critical model whose ground state breaks conformal invariance such as in those described by non unitary conformal field theories behaves as ceff n 1 2n log l where ceff c 24delta 0 is the effective central charge c which may be negative is the central charge of the conformal field theory and delta 0 is the lowest holomorphic conformal dimension in the theory we also obtain results for models with boundaries and with a large but finite correlation length and we show that if the lowest conformal eigenspace is logarithmic then there is an additional term proportional to log log l these results generalize the well known expressions for unitary models we provide a general proof and report on numerical evidence for a non unitary spin chain and an analytical computation using the corner transfer matrix method for a non unitary lattice model we use a new algebraic technique for studying the branching that arises within the replica approach and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non unitary models

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

scaling dimensions and conformal anomaly in anisotropic lattice spin models the effect of anisotropic interactions on the eigenvalue spectrum of the row to row transfer matrix of critical lattice spin models is investigated it is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta ay ax sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy for exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices this is done for the eight vertex model hard hexagons and interacting hard squares and it is found that theta pi u lambda 10u 3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter for each of these models the amplitude of the finite size corrections to the free energy at criticality is found to be of the form pi zeta c 6n2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c 1 4 5 and 7 10 respectively this is an analytic result for the eight vertex model for the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to n 48 finally the authors argue that the anisotropy angle for magnetic hard squares and the q state potts models is also given by theta pi u lambda so this result is quite general

some finite size amplitudes and critical exponents for potts and ashkin teller quantum chains analytic expressions are derived for the leading finite size corrections to a class of energy eigenvalues of the potts and ashkin teller quantum chains making use of their equivalences with a modified xxz heisenberg chain assuming conformal invariance exact results are thence obtained for some bulk scaling dimensions and surface exponents in these models

the spectral gap and the dynamical critical exponent of an exact solvable probabilistic cellular automaton abstract we obtained the exact solution of a probabilistic cellular automaton related to the diagonal to diagonal transfer matrix of the six vertex model on a square lattice the model describes the flow of ants or particles traveling on a one dimensional lattice whose sites are small craters containing sleeping or awake ants two kinds of particles we found the bethe ansatz equations and the spectral gap for the time evolution operator of the cellular automaton from the spectral gap we show that in the asymmetric case it belongs to the kardar parisi zhang kpz universality class exhibiting a dynamical critical exponent value z 3 2 this result is also obtained from a direct monte carlo simulation by evaluating the lattice size dependence of the decay time to the stationary state

stochastic higher spin vertex models on the line we introduce a four parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of bethe ansatz eigenfunctions and which enjoy certain markov dualities using this for the systems started in step initial data we write down nested contour integral formulas for moments and fredholm determinant formulas for laplace type transforms taking various choices or limits of parameters this family degenerates to many of the known exactly solvable models in the kardar parisi zhang universality class as well as leads to many new examples of such models in particular asymmetric simple exclusion process the stochastic six vertex model q totally asymmetric simple exclusion process and various directed polymer models all arise in this manner our systems are constructed from stochastic versions of the r matrix related to the six vertex model one of the key tools used here is the fusion of r matrices and we provide a probabilistic proof of this procedure

time dependent correlation functions in a one dimensional asymmetric exclusion process we study a one dimensional anisotropic exclusion process describing particles injected at the origin moving to the right on a chain of l sites and being removed at the right boundary we construct the steady state and compute the density profile exact expressions for all equal time n point density correlation functions and the time dependent two point function in the steady state as functions of the injection and absorption rates we determine the phase diagram of the model and compare our results with predictions from dynamical scaling and discuss some conjectures for other exclusion models

four state models and clifford algebras with appropriate boundary conditions the anisotropic xy chain in a magnetic field is known to be invariant under quantum group transformations we generalize this model defining a class of integrable chains with several fermionic degrees of freedom per site in order to maintain the quantum group symmetry a general condition on the parameters of these systems is derived it is shown that the corresponding quantum algebra is a multi parameter deformation of the clifford algebra discussing a special physical example we observe a new type of zero mode

wulff construction and anisotropic surface properties of two dimensional eden clusters a relation between the shape of eden clusters and the number of perimeter sites per unit area of the surface is derived which is analogous to the wulff construction of equilibrium shapes in thermodynamic systems new data are presented for the surface width and the surface skewness of eden clusters grown on a square lattice the width depends on the average orientation of the surface with respect to the underlying lattice its corrections to scaling are discussed the skewness has unexpected changes of sign

roughening of the interfaces in 1 1 dimensional two component surface growth with an admixture of random deposition we simulate competitive two component growth on a one dimensional substrate of l sites one component is a poisson type deposition that generates kardar parisi zhang kpz correlations the other is random deposition rd we derive the universal scaling function of the interface width for this model and show that the rd admixture acts as a dilatation mechanism to the fundamental time and height scales but leaves the kpz correlations intact this observation is generalized to other growth models it is shown that the flat substrate initial condition is responsible for the existence of an early nonscaling phase in the interface evolution the length of this initial phase is a nonuniversal parameter but its presence is universal we introduce a method to measure the length of this initial nonscaling phase in application to parallel and distributed computations the important consequence of the derived scaling is the existence of the upper bound for the desynchronization in a conservative update algorithm for parallel discrete event simulations it is shown that such algorithms are generally scalable in a ring communication topology

totally asymmetric exclusion process with extended objects a model for protein synthesis the process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent covering more than one lattice site we expand the well studied totally asymmetric exclusion process in which particles typically cover a single lattice site to include cases with extended objects exact solutions can be determined for a uniform closed system we analyze the uniform open system through two approaches first a continuum limit produces a modified diffusion equation for particle density profiles second an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase finally we briefly consider approximate approaches to a nonuniform open system with quenched disorder in the particle hopping rates and compare these approaches with monte carlo simulations

finite size scaling and the toroidal partition function of the critical asymmetric six vertex model finite size corrections to the energy levels of the asymmetric six vertex model transfer matrix are considered using the bethe ansatz solution for the critical region the non universal complex anisotropy factor is related to the bulk susceptibilities the universal gaussian coupling constant g is also related to the bulk susceptibilities as g 2h 1 2 pi h being the hessian of the bulk free energy surface viewed as a function of the two fields the modular covariant toroidal partition function is derived in the form of the modified coulombic partition function which embodies the effect of incommensurability through two mismatch parameters the effect of twisted boundary conditions is also considered

totally asymmetric exclusion processes with particles of arbitrary size the steady state currents and densities of a one dimensional totally asymmetric exclusion process tasep with particles that occlude an integer number d of lattice sites are computed using various mean field approximations and monte carlo simulations taseps featuring particles of arbitrary size are relevant for modelling systems such as mrna translation vesicle locomotion along microtubules and protein sliding along dna we conjecture that the nonequilibrium steady state properties separate into low density high density and maximal current phases similar to those of the standard d 1 tasep a simple mean field approximation for steady state particle currents and densities is found to be inaccurate however we find local equilibrium particle distributions derived from a discrete tonks gas partition function yield apparently exact currents within the maximal current phase for the boundary limited phases the equilibrium tonks gas distribution cannot be used to predict currents phase boundaries or the order of the phase transitions however we employ a refined mean field approach to find apparently exact expressions for the steady state currents boundary densities and phase diagrams of the d 1 tasep extensive monte carlo simulations are performed to support our analytic mean field results

structural stability of vector fields on 3 manifolds with boundary abstract from the structural stability viewpoint vector fields on m which are tangent to the boundary of m are analysed a class of vector fields is characterized as structurally stable this class corresponds to the class of morse smale vector fields on closed manifolds studied by palis peixoto smale and others in this class new phenomena occur as saddle connections along the boundary of m which are persistent by small perturbations thus the techniques introduced by j palis topology 8 1969 which inspired the proof of the stability of a vector field in such a class were substantially changed such modifications represent a main difficulty in extending our results to higher dimensions

relaxation spectrum of the asymmetric exclusion process with open boundaries we calculate numerically the exact relaxation spectrum of the totally asymmetric simple exclusion process tasep with open boundary conditions on lattices up to 16 sites in the low and high density phases and along the nonequilibrium first order phase transition between these phases but sufficiently far away from the second order phase transition into the maximal current phase the low lying spectrum corresponding to the longest relaxation times agrees well with the spectrum of a biased random walker confined to a finite lattice of the same size the hopping rates of this random walk are given by the hopping rates of a shock a domain wall separating stationary low and high density regions which are calculated in the framework of a recently developed non equilibrium version of zel dovich s theory of the kinetics of first order transitions we conclude that the description of the domain wall motion in the tasep in terms of this theory of boundary induced phase transitions is meaningful for very small systems of the order of ten lattice sites

fluctuations for stationary q tasep we consider the q totally asymmetric simple exclusion process q tasep in the stationary regime and study the fluctuation of the position of a particle we first observe that the problem can be studied as a limiting case of an n particle q tasep with a random initial condition and with particle dependent hopping rate then we explain how this n particle q tasep can be encoded in a dynamics on a two sided gelfand tsetlin cone described by a two sided q whittaker process and present a fredholm determinant formula for the q laplace transform of the position of a particle two main ingredients in its derivation is the ramanujan s bilateral summation formula and the cauchy determinant identity for the theta function with an extra parameter based on this we establish that the position of a particle obeys the universal stationary kpz distribution the baik rains distribution in the long time limit

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

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

exclusion process for particles of arbitrary extension hydrodynamic limit and algebraic properties the behaviour of extended particles with exclusion interaction on a one dimensional lattice is investigated the basic model is called l asep as a generalization of the asymmetric exclusion process asep to particles of arbitrary length l stationary and dynamical properties of the l asep with periodic boundary conditions are derived in the hydrodynamic limit from microscopic properties of the underlying stochastic many body system in particular the hydrodynamic equation for the local density evolution and the time dependent diffusion constant of a tracer particle are calculated as a fundamental algebraic property of the symmetric exclusion process the su 2 symmetry is generalized to the case of extended particles

2 equivariant vector fields on 3 manifolds moduli of stability and genericity an open and dense class of vector fields on 3 dimensional compact manifolds equivariant under the action of s0 2 is defined each such vector field has finite moduli of stability we also exhibit an open and dense subset of the s0 2 equivariant gradient vector fields which are structurally stable introduction and statement of results let g so 2 be the lie group of rotations acting smoothly and effectively on a smooth three dimensional closed compact connected boundaryless manifold m let xr g m denote the space of all g equivariant c vector field endowed with the cr topology in this paper we define a class of vector fields in xr g m named g morse smale which is open and dense and prove that each vector field in this class has finite moduli of stability we also make a similar study for the corresponding class in the c g equivariant gradient vector fields in this setting we actually have structural stability three main notions play a role in our work a finite moduli of stability p2 it means that we can parametrize a neighborhood of the vector field by a finite number of n parameter families of vector fields b normal hyperbolicity c the third ingredient is g transversality of stable and unstable manifolds f2 bi in our case this concept coincides with that of stratumwise transversality the compactness of g implies that m is finitely stratified by g invariant submanifolds with each stratum invariant by g equivariant vector fields stratumwise transversality is the usual transversality inside each of these strata now we state our results in a more precise way definitions may be found on 1 a g equivariant vector field on m is said to be g morse smale if i its nonwandering set consists of a finite number of normally hyperbolic critical elements ii the stable and unstable manifolds of critical elements are g transversal as it was said before in the following theorems g is so 2 and dim m 3 theorem a the set of g morse smale vector fields is open in xr g m theorem b the set of g morse smale vector fields is dense in xr g m if r 1 if r 2 and m g is orientable the projective plane the klein bottle or received by the editors march 31 1986 and in revised form may 5 1987 1980 mathematics subject classification primary 58f10

diagonalization of the infinite q boson system we present a hierarchy of commuting operators in fock space con taining the q boson hamiltonian on z and show that the operators in question are simultaneously diagonalized by hall littlewood functions as an applica tion the n particle scattering operator is computed the q boson model constitutes a one dimensional exactly solvable particle system in fock space bik based on the q oscillator algebra ks ch 5 in the case of periodic boundary conditions i e with particles hopping on the finite lattice zm the integrability the spectrum and the eigenfunctions of the hamiltonian were analyzed by means of the algebraic bethe ansatz method bik remarkably these eigenfunctions turn out to be hall littlewood functions t k cf also j for an alternative construction of hall littlewood functions in fock space based on deformed vertex operator algebras with applications in the study of kp functions arising from generating functions of weighted plane partitions fw with the aid of explicit expressions for the commuting quantum integrals arising from an infinite dimensional solution of the yang baxter equation it was very recently demonstrated k that the eigenvalue problem for the q boson system on zm is in fact equivalent to that of an integrable discretization d of the celebrated delta bose gas on the circle ll the present work addresses the spectral problem and the integrability of the q boson system on the infinite lattice z specifically we demonstrate that the eigenfunctions of this infinite q boson system are again given by hall littlewood functions and provide explicit formulas for a complete hierarchy of operators com muting with the hamiltonian these formulas are natural infinite dimensional ana logues of the above mentioned expressions in k for the finite q boson system on zm finally the n particle scattering operator is computed as an application of ruijsenaars general scattering results in r2 2 the infinite q boson system

equivariant diffeomorphisms with simple recurrences on two manifolds we consider the class of diffeomoiphisms on compact two dimensional manifolds which are invanant under the action of a compact lie group c and whose nonwandering set consists of a finite number of g orbits we descnbe the modulus of stability of almost all diffeomorphisms in this class introduction and statement of results let g be a compact lie group acting smoothly on a smooth closed connected two dimensional manifold m we say that a g equivariant diffeomorphism of m has simple recurrences if its nonwandering set consists of a finite number of normally hyperbolic critical elements denote bysthe set of g equivariant diffeomorphisms with simple recurrences whose stable and unstable manifolds of its critical elements are g transversal there are many diffeomorphisms in swhich are not structurally stable however for some of them it may be possible to parametrize all the topological conjugacy classes in their neighborhood by finitely many real parameters in this case we say that they have finite modulus of stability otherwise their modulus of stability is infinite in this paper we study the modulus of stability of diffeomorphisms in a dense set cs now we will state our result in a precise way a c diffeomorphismf m m is g equivariant if f g x g f x for all x e m and g e g the set of such diffeomorphisms with the cx topology is denoted by diffg m an element t in diffg m induces a g x z action on m given by g n x g tn x a critical element of t is a compact g x z orbit a critical element a of f is normally hyperbolic if there is a continuous invariant splitting of the tangent bundle of m over a tam to es e3 eu and constants c 0 0 o the stable manifold of a ws a y e m distance offn y to a tends to zero when n oo has g t invariant foliation gss f whose leaves are the strong stable manifolds of the points in a similarly the unstable manifolds of a have a g f invariant foliation sauu f here in dimension two we have that a is either a periodic orbit or a finite union of circles in the first case a is just a hyperbolic orbit and soss f eau f is a finite number of submanifolds in the second case a is an received by the editors april 13 1984 and in revised form august 21 1984 1980 muthematics s bject clczssification primary 58f10 secondary 57s15 kez words clnd phrclses compact lie groups equivariant diffeomorphisms structural stability modulus of stability l the second and third authors were partially supported by cnpq brazil c 1985 american mathematical society 0002 9947 85 1 00 25 per page

a bayesian semiparametric model for case control studies with errors in variables summary we develop a model and a numerical estimation scheme for a bayesian approach to inference in case control studies with errors in covariables the model proposed in this paper is based on a nonparametric model for the unknown joint distribution for the missing data the observed covariates and the proxy this nonparametric distribution defines the measurement error component of the model which relates the missing covariates x with a proxy w the oxymoron nonparametric bayes refers to a class of flexible mixture distributions for the likelihood of disease given covariates we choose a logistic regression model by using a parametric disease model and nonparametric exposure model we obtain robust interpretable results quantifying the effect of exposure

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

structural stability of equivariant vector fields on two manifolds a class of vector fields on two dimensional manifolds equivariant under the action of a compact lie group is defined properties of openness structural ability and density are proved introduction and statement of results let g be a compact lie group acting smoothly on a smooth compact connected two dimensional manifold m in this paper we define a subset of the space rg m of c r 1 equivariant vector fields on m which correspond to the morse smale vector fields studied by peixoto palis smale 17 11 15 and others see 13 we prove that this subset of equivariant vector fields which we call g morse smale vector fields is open and each gmorse smale vector field x is equivariantly structurally stable in rc m that is if y e irc m is near x then there exists an equivariant homeomorphism of m that sends trajectories of x into trajectories of y also we prove that it is dense in rc m with some exceptions the most important exception corresponds to the fact that the density of the morse smale vector fields for nonorientable 2 manifolds in the cr topology with r s 2 is still an open question see gutierrez 6 our results generalize those of peixoto 17 the definition of g morse smale vector fields given here is different from that given by field 3 5 in that it allows the presence of graphs in the nonwandering set in this way we have enough vector fields to get density along with structural stability however in higher dimensions the situation is unclear since no results on structural stability have been proved so far we believe that the concept of modulus of stability melo 10 palis 12 seems to be more appropriate to the equivariant framework examples of equivariant vector fields on 3 manifolds with modulus of stability equal to one are given in 14 in fact this phenomenom appears even locally a g equivariant vector field x on m defines a g x r action on m see 1 a critical element of x is a compact g x r orbit we require that it be normally hyperbolic of course normal hyperbolicity does not imply local structural stability but it may provide modulus of stability we may impose an additional condition on the g action see 1 2 in order to get local received by the editors february 24 1982 and in revised form may 24 1983 1980 mathematics subject classification primary 58f10 secondary 57e1s

solvable reaction diffusion processes without exclusion for reaction diffusion processes without exclusion in which the particles can exist in the same site of a one dimensional lattice we study all the integrable models which can be obtained by imposing a boundary condition on the master equation of the asymmetric diffusion process the annihilation process is also added the bethe ansatz solution and the exact n particle conditional probabilities are obtained

pairwise balance and invariant measures for generalized exclusion processes we characterize the steady state of a driven diffusive lattice gas in which each site holds several particles and the dynamics is activated and asymmetric using a quantum hamiltonian formalism we show that for arbitrary transition rates the model has product invariant measure in the steady state a pairwise balance condition is shown to hold configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to pairwise balance is more general than the condition of detailed balance and holds in the non equilibrium steady state of a number of stochastic models

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

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