a sufficient criterion for integrability of stochastic many body dynamics and quantum spin chains we propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded volume interaction and the quantum mechanics of the associated quantum spin chains respectively the time dependent algebra which is obtained from the action of the markov generator of the exclusion process or quantum hamiltonian of the spin chain respectively is given in terms of a set of quadratic relations by analysing the permutation consistency of the induced cubic relations we obtain sufficient conditions on the hopping rates i e the quantum mechanical interaction constants which allow us to identify integrable models from the dynamical algebra we construct the quadratic algebra of zamolodchikov type the associativity of which is a yang baxter equation we also obtain directly from the dynamical matrix product ansatz the bethe ansatz equations for the spectra of these models

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

a discrete nodal domain theorem for trees let g be a connected graph with n vertices and let x x1 xn be a real vector a positive negative sign graph of the vector x is a maximal connected subgraph of g on vertices xi 0 xi 0 for an eigenvalue of a generalized laplacian of a tree we characterize the maximal number of sign graphs of an eigenvector we give an o n2 time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an np complete problem author s abstract

the asymmetric exclusion process and brownian excursions we consider the totally asymmetric exclusion process tasep in one dimension in its maximal current phase we show by an exact calculation that the non gaussian part of the fluctuations of density can be described in terms of the statistical properties of a brownian excursion numerical simulations indicate that the description in terms of a brownian excursion remains valid for more general one dimensional driven systems in their maximal current phase

asymmetric simple exclusion process with open boundaries and askey wilson polynomials we study the one dimensional asymmetric simple exclusion process asep with open boundary conditions particles are injected and ejected at both boundaries it is clarified that the steady state of the model is intimately related to the askey wilson polynomials the partition function and the n point functions are obtained in the integral form with four boundary parameters the thermodynamic current is evaluated to confirm the conjectured phase diagram

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

minimum vertex covers and the spectrum of the normalized laplacian on trees abstract we show that in the graph spectrum of the normalized graph laplacian on trees the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers in particular for the eigenvalue 1 its multiplicity is related to the size of a minimum vertex cover and zero entries of its eigenvectors correspond to vertices in minimum vertex covers while for eigenvalues near 1 their distance to 1 can be estimated from minimum vertex covers and for the largest eigenvalue smaller than 1 the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives

exact results for 2d directed animals on a strip of finite width proves a conjecture giving the exact number of directed animals of s sites with any root on a strip of finite width of a square lattice the authors also rederive more simply some previous results concerning the connective constant and particular eigenvectors of the transfer matrix

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

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

on matrix product ansatz for asymmetric simple exclusion process with open boundary in the singular case we study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the singular case alpha beta q n gamma delta when the standard form of the matrix product ansatz of derrida et al j phys a 26 7 1493 1517 1993 does not apply in our approach the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra one of the functionals varphi 1 is defined on the entire algebra and determines stationary probabilities for large systems on l ge n 1 sites the other functional varphi 0 is defined only on a finite dimensional linear subspace of the algebra and determines stationary probabilities for small systems on l n 1 sites functional varphi 0 vanishes on non constant askey wilson polynomials and in non singular case becomes an orthogonality functional for the askey wilson polynomials

characteristics polynomial of normalized laplacian for trees here we find the characteristics polynomial of normalized laplacian of a tree the coefficients of this polynomial are expressed by the higher order general randic indices for matching whose values depend on the structure of the tree we also find the expression of these indices for starlike tree and a double starlike tree hm p q moreover we show that two cospectral hm p q of the same diameter are isomorphic

matrix product ansatz as a tridiagonal algebra in the matrix product states approach to interacting multiparticle systems the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process the states involved in the matrix elements are determined by the boundary conditions this reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates led by the importance of the boundary conditions we consider the boundary operators as generators of a tridiagonal algebra whose irreducible modules are the askey wilson polynomials the matrices of the matrix product ansatz obey the tridiagonal algebraic relations as well for particular values of the structure constants this suggests the formulation of the steady state properties in terms of noncommutative matrices generating a tridiagonal askey wilson algebra the previously known representations both infinite dimensional and finite dimensional ones are recovered within the tridiagonal framework

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

the application of the yang lee theory to study a phase transition in a non equilibrium system we study a phase transition in a non equilibrium model first introduced in ref 5 using the yang lee description of equilibrium phase transitions in terms of both canonical and grand canonical partition function zeros the model consists of two different classes of particles hopping in opposite directions on a ring on the complex plane of the diffusion rate we find two regions of analyticity for the canonical partition function of this model which can be identified by two different phases the exact expressions for both distribution of the canonical partition function zeros and their density are obtained in the thermodynamic limit the fact that the model undergoes a second order phase transition at the critical point is confirmed we have also obtained the grand canonical partition function zeros of our model numerically the similarities between the phase transition in this model and the bose einstein condensation has also been studied

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

exact results for the asymmetric simple exclusion process with a blockage we present new results for the current as a function of transmission rate in the one dimensional totally asymmetric simple exclusion process tasep with a blockage that lowers the jump rate at one site from one tor 1 exact finitevolume results serve to bound the allowed values for the current in the infinite system this proves the existence of a nonequilibrium phase transition corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state a series expansion inr derived from the finite systems is proven to be asymptotic for all sufficiently large systems pade approximants based on this series which make specific assumptions about the nature of the singularity atr 1 match numerical data for the infinite system to 1 part in 104

multiple shocks in a driven diffusive system with two species of particles abstract a one dimensional driven diffusive system with two types of particles and nearest neighbors interactions has been considered on a finite lattice with open boundaries the particles can enter and leave the system from both ends of the lattice and there is also a probability for converting the particle type at the boundaries we will show that on a special manifold in the parameters space multiple shocks evolve in the system for both species of particles which perform continuous time random walks on the lattice

exact solution of a cellular automaton for traffic we present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions e g cars can enter and leave a part of a highway with certain probabilities the model studied is the asymmetric exclusion process asep with simultaneous updating of all sites it is equivalent to a special case vmax 1 of the nagel schreckenberg model for highway traffic which has found many applications in real time traffic simulations the simultaneous updating induces additional strong short range correlations compared to other updating schemes the stationary state is written in terms of a matrix product solution the corresponding algebra which expresses a system size recursion relation for the weights of the configurations is quartic in contrast to previous cases in which the algebra is quadratic we derive the phase diagram and compute various properties such as density profiles two point functions and the fluctuations in the number of particles cars in the system the current and the density profiles can be mapped onto the asep with other time discrete updating procedures through use of this mapping our results also give new results for these models

darboux and binary darboux transformations for the nonautonomous discrete kp equation it is shown how darboux and binary darboux transformations for a nonautonomous discrete kp equation can be obtained from fermion analysis this equation is obtained by considering a generalized miwa transformation it is also shown to be linked to the discrete kp equation by a special gauge transformation the darboux and binary darboux transformations are used to discuss general classes of solutions in the form of casorati and gramm type determinants n soliton solutions are discussed as well

unitary representations of uq 2 ℝ the modular double and the multiparticle q deformed toda chain the paper deals with the analytic theory of the quantum q deformed toda chains the technique used combines the methods of representation theory and the quantum inverse scattering method the key phenomenon which is under scrutiny is the role of the modular duality concept first discovered by l faddeev in the representation theory of noncompact semisimple quantum groups explicit formulae for the whittaker vectors are presented in terms of the double sine functions and the wave functions of the n particle q deformed open toda chain are given as a multiple integral of the mellin barnes type for the periodic chain the two dual baxter equations are derived

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

partially asymmetric simple exclusion model in the presence of an impurity on a ring we study a generalized two species model on a ring the original model mallick k 1996 j phys a math gen 29 5375 describes ordinary particles hopping exclusively in one direction in the presence of an impurity the impurity hops with a rate different from that of ordinary particles and can be overtaken by them here we let the ordinary particles also hop backward with rate q using the matrix product ansatz we obtain the relevant quadratic algebra a finite dimensional representation of this algebra enables us to compute the stationary bulk density of the ordinary particles as well as the speed of impurity on a set of special surfaces of the parameter space we will obtain the phase structure of this model in the accessible region and show how the phase structure of the original model is modified in the infinite volume limit this model presents a shock in one of its phases

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

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

two way traffic flow exactly solvable model of traffic jam we study completely asymmetric two channel exclusion processes in one dimension it describes a two way traffic flow with cars moving in opposite directions the interchannel interaction makes cars slow down in the vicinity of approaching cars in the other lane particularly we consider in detail the system with a finite density of cars on one lane and a single car on the other when the interchannel interaction reaches a critical value a traffic jam occurs which turns out to be of first order phase transition we derive exact expressions for the average velocities the current the density profile and the k point density correlation functions we also obtain the exact probability of two cars being in one lane of distance r apart provided there is a finite density of cars on the other lane and show that the two cars form a weakly bound state in the jammed phase

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

construction of a matrix product stationary state from solutions of a finite size system stationary states of stochastic models which have n states per site in matrix product form are considered first we give a necessary condition for the existence of a finite m dimensional matrix product stationary state for any n m second we give a method to construct the matrices from the stationary

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

matrix coordinate bethe ansatz applications to xxz and asep models we present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process asep and xxz models with special constraints on the boundaries the method combines both recent constructions of coordinate bethe ansatz and the old method of matrix ansatz specific to the asep this matrix coordinate bethe ansatz can be viewed as a non commutative coordinate bethe ansatz the non commutative part being related to the algebra appearing in the matrix ansatz

a matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process we study the fluctuations of the position of an impurity in the asymmetric exclusion process on a ring with an arbitrary number of particles and holes the steady state of this model is exactly known and four different phases appear in the limit of a large system we calculate the diffusion constant of the impurity by using a matrix product method and also obtain a representation for unequal time correlation functions we show that our results found by the matrix ansatz agree with those obtained previously by the bethe ansatz

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

quadratic harnesses q commutations and orthogonal martingale polynomials we introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a commutation relation this implies that quadratic harnesses are essentially determined uniquely by five numerical constants explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest

partially asymmetric exclusion process with open boundaries exclusive diffusion on a one dimensional lattice is studied in the model particles hop stochastically in both directions but with different rates at the ends of the lattice particles are injected and removed the exact stationary probability measure is represented in the form of a matrix product as a generalization of the solution given by derrida et al j phys a 26 1493 1993 for the fully asymmetric process the phase diagram of the current on the infinite lattice is obtained analytic expressions for the current in the different phases are derived the model is equivalent to an xxz heisenberg chain with a certain type of boundary terms the ground state of which corresponds to the stationary solution of the master equation

spectral classes of regular random and empirical graphs abstract we define a pseudo distance between graphs based on the spectrum of the normalized laplacian since this quantity can be computed easily or at numerically estimated it is suitable for comparing in particular large graphs numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity the asymptotic behavior of the laplacian spectrum furthermore yields a tool for classifying families of graphs in such a way that the distance of two graphs from the same family is bounded by o 1 n in terms of size n of their vertex sets

multiple transit paths and density correlation functions in partially asymmetric simple exclusion process we consider the partially asymmetric simple exclusion process pasep when its steady state probability distribution function can be written in terms of a linear superposition of product measures with a finite number of shocks in this case the pasep can be mapped into an equilibrium walk model defined on a diagonally rotated square lattice in which each path of the walk model has several transits with the horizontal axis we show that the multiple point density correlation function in the pasep is related to the probability that a path has multiple contacts with the horizontal axis from the above or below

disordered exclusion process revisited some exact results in the low current regime we study the steady state of the totally asymmetric simple exclusion process with inhomogeneous hopping rates associated with sites site wise disorder using the fact that the non normalized steady state weights which solve the master equation are polynomials in all the hopping rates we propose a general method for calculating their first few lowest coefficients exactly in the case of binary disorder where all slow sites share the same hopping rate r 1 we apply this method to calculate the steady state current up to the quadratic term in r for some particular disorder configurations for the most general non binary disorder we show that in the low current regime the current is determined solely by the current minimizing subset of equal hopping rates provided all other slow rates are much larger our approach can be readily applied to any other driven diffusive system with unidirectional hopping if one can identify a hopping rate such that the current vanishes when this rate is set to zero

calculating the normalized laplacian spectrum and the number of spanning trees of linear pentagonal chains abstract let w n be a linear pentagonal chain with 2 n pentagons in this article according to the decomposition theorem for the normalized laplacian polynomial of w n we obtain that the normalized laplacian spectrum of w n consists of the eigenvalues of two special matrices l a of order 3 n 1 and l s of order 2 n 1 together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices explicit closed form formulas for the degree kirchhoff index and the total number of spanning trees of w n are derived respectively finally it is interesting to find that the degree kirchhoff index of w n is approximately one half of its gutman index

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

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

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

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

finite dimensional representations of a shock algebra the algebra describing a shock measure in the asymmetric simple exclusion model seen from a second class particle has finite dimensional representations if and only if the asymmetry parameterp of the model and the left and right asymptotic densitiesp of the shock satisfy 1 p p r p 1 p p 1 p for some integerr 1 the minimal dimension of the representation is then 2r these representations can be used to calculate correlation functions in the model

continued fractions and the partially asymmetric exclusion process we note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process pasep lends itself to interpretation as the transfer matrix for weighted motzkin lattice paths a continued fraction j fraction representation of the lattice path generating function is particularly well suited to discussing the pasep for which the paths have height dependent weights we show that this not only allows a succinct derivation of the normalization and correlation lengths of the pasep but also reveals how finite dimensional representations of the pasep algebra valid only along special lines in the phase diagram relate to the general solution that requires an infinite dimensional representation

random vector and time series definition and synthesis from matrix product representations from statistical physics to hidden markov models inspired from modern out of equilibrium statistical physics models a matrix product based framework is defined and studied that permits the formal definition of random vectors and time series whose desired joint distributions are a priori prescribed its key feature consists of preserving the writing of the joint distribution as the simple product structure it has under independence while inputing controlled dependencies amongst components this is obtained by replacing the product of probability densities by a product of matrices of probability densities it is first shown that this matrix product model can be remapped onto the framework of hidden markov models second combining this dual perspective enables us both to study the statistical properties of this model in terms of marginal distributions and dependencies a stationarity condition is notably devised and to devise an efficient and accurate numerical synthesis procedure a design procedure is also described that permits the tuning of model parameters to attain targeted statistical properties pedagogical well chosen examples of times series and multivariate vectors aim at illustrating the power and versatility of the proposed approach and at showing how targeted statistical properties can actually be prescribed

the normalized laplacians on both k triangle graph and k quadrilateral graph with their applications abstract the k triangle graph tk g is obtained from a graph g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 2 whereas the k quadrilateral graph qk g is obtained from g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 3 in this paper we completely determine the normalized laplacian spectrum on tk g resp qk g for any connected graph g k 2 as applications the correlation between the degree kirchhoff index the kemeny s constant and the number of spanning trees of tk g resp qk g the r th iterative k triangle graph t r k g the r th iterative k quadrilateral graph q r k g and those of g are derived our results extend those main results obtained in xie et al 2016 and li and hou 2017

the normalized laplacian spectrum of quadrilateral graphs and its applications abstract the quadrilateral graph q g of g is obtained from g by replacing each edge in g with two parallel paths of lengths 1 and 3 in this paper we completely describe the normalized laplacian spectrum on q g for any graph g as applications the significant formulae to calculate the multiplicative degree kirchhoff index the kemeny s constant and the number of spanning trees of q g and the quadrilateral iterative graph q r g are derived

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

some general features of matrix product states in stochastic systems we will prove certain general relations in the matrix product ansatz mpa for one dimensional stochastic systems which are true in both random and sequential updates we will derive general mpa expressions for the currents and current correlators and find the conditions in the mpa formalism under which the correlators are site independent or completely vanishing

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

on the spectrum of the normalized graph laplacian we investigate how the spectrum of the normalized geometric graph laplacian is affected by operations like motif dougling graph splitting or joining the multiplicity of the eigenvalue 1 or equivalently the dimension of the kernel of the adjacency matrix of the graph is of particular interest this multiplicity can be increased for instance by motif doubling

generalized bethe ansatz solution of a one dimensional asymmetric exclusion process on a ring with blockage we present a model for a one dimensional anisotropic exclusion process describing particles moving deterministically on a ring of lengthl with a single defect across which they move with probability 0 p 1 this model is equivalent to a two dimensional six vertex model in an extreme anisotropic limit with a defect line interpolating between open and periodic boundary conditions we solve this model with a bethe ansatz generalized to this kind of boundary condition we discuss in detail the steady state and derive exact expressions for the currentj the density profilen x and the two point density correlation function in the thermodynamic limitl the phase diagram shows three phases a low density phase a coexistence phase and a high density phase related to the low density phase by a particle hole symmetry in the low density phase the density profile decays exponentially with the distance from the boundary to its bulk value on a length scale on the phase transition line diverges and the currentj approaches its critical valuejc p as a power law jc j 1 2 in the coexistence phase the width of the interface between the high density region and the low density region is proportional tol1 2 if the density f 1 2 and 0 independent ofl if 1 2 the connected two point correlation function turns out to be of a scaling form with a space dependent amplitude n x1 x2 a x2 ake r withr x2 x1 and a critical exponent 0

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

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

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

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

multi species asymmetric exclusion process in ordered sequential update a multi species generalization of the asymmetric simple exclusion process is studied with ordered sequential and sub lattice parallel updating schemes in this model particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability using the matrix product ansatz technique we obtain the relevant algebra and study the uncorrelated stationary state of the model both for an open system and on a ring a complete comparison between the physical results in these updates and those of random sequential introduced in karimipour v 1999 a multi species asep and its relation to traffic flow phys rev e 59 205 and karimipour v 1998 a multi species asep steady state and correlation functions on a periodic lattice preprint cond mat 9809193 is made

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

matrix product states of three families of one dimensional interacting particle systems abstract the steady states of three families of one dimensional non equilibrium models with open boundaries first proposed in n j phys 5 2003 145 1 are studied using a matrix product formalism it is shown that their associated quadratic algebras have two dimensional representations provided that the transition rates lie on specific manifolds of parameters exact expressions for the correlation functions of each model have also been obtained we have also studied the steady state properties of one of these models first introduced in j phys a math gen a 36 2003 74 with more details by introducing a canonical ensemble we calculate the canonical partition function of this model exactly using the yang lee theory of phase transitions we spot a second order phase transition from a power law to a jammed phase the density profile of particles in each phase has also been studied a simple generalization of this model in which both the left and the right boundaries are open has also been introduced it is shown that double shock structures may evolve in the system under certain conditions

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

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

fusion procedure for open chains the authors have generalized sklyanin s approach of constructing open integrable quantum spin chains to the case of pt invariant r matrices they formulate a fusion procedure for such chains in particular they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated applications of these results include constructing open integrable higher spin chains as well as obtaining functional equations for transfer matrix eigenvalues which may be solved by an analytical bethe ansatz

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

q special functions with q 1 and their application to discrete integrable systems solutions for certain discrete integrable systems are constructed by using integral solutions for hypergeometric q difference systems with q 1 we apply a solution for lauricella s d type hypergeometric q difference system with q 1 to construct a solution for the discrete kp hierarchy furthermore an integral solution of a q difference analogue of bessel s equation is newly introduced and applied to construct a solution for a q difference analogue of the cylindrical toda equation with q 1

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

from quantum affine symmetry to the boundary askey wilson algebra and the reflection equation within the quantum affine algebra representation theory we construct linear covariant operators that generate the askey wilson algebra it has the property of a coideal subalgebra which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk the generators of the askey wilson algebra are implemented to construct an operator valued k matrix a solution of a spectral dependent reflection equation we consider the open driven diffusive system where the askey wilson algebra arises as a boundary symmetry and can be used for an exact solution of the model in the stationary state we discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the askey wilson algebra to the reflection equation

weakly coupled antiparallel totally asymmetric simple exclusion processes we study a system composed of two parallel totally asymmetric simple exclusion processes with open boundaries where the particles move in the two lanes in opposite directions and are allowed to jump to the other lane with rates inversely proportional to the length of the system stationary density profiles are determined and the phase diagram of the model is constructed in the hydrodynamic limit by solving the differential equations describing the steady state of the system analytically for vanishing total current and numerically for nonzero total current the system possesses phases with a localized shock in the density profile in one of the lanes similarly to exclusion processes endowed with nonconserving kinetics in the bulk besides the system undergoes a discontinuous phase transition where coherently moving delocalized shocks emerge in both lanes and the fluctuation of the global density is described by an unbiased random walk this phenomenon is analogous to the phase coexistence observed at the coexistence line of the totally asymmetric simple exclusion process however as a consequence of the interaction between lanes the density profiles are deformed and in the case of asymmetric lane change the motion of the shocks is confined to a limited domain

combinatorics of the two species asep and koornwinder moments abstract in previous work 12 13 14 the first and third authors introduced staircase tableaux which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process asep and for the moments of the askey wilson weight function the fact that the asep and askey wilson moments are related at all is unexpected and is due to 45 the asep is a model of particles hopping on a one dimensional lattice of n sites with open boundaries particles can enter and exit at both left and right borders it was introduced around 1970 34 43 and is cited as a model for both traffic flow and translation in protein synthesis meanwhile the askey wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials so from this previous work we have the relationship asep staircase tableaux askey wilson moments the askey wilson polynomials can be viewed as the one variable case of the multivariate koornwinder polynomials which are also known as the macdonald polynomials attached to the non reduced affine root system c n c n it is natural then to ask whether one can generalize the relationships among the asep askey wilson moments and staircase tableaux in such a way that koornwinder moments replace askey wilson moments in 15 we made a precise link between koornwinder moments and the two species asep a generalization of the asep which has two species of particles with different weights in this article we introduce rhombic staircase tableaux and show that we have the relationship 2 species asep rhombic staircase tableaux koornwinder moments in particular we give formulas for the stationary distribution of the two species asep and for koornwinder moments in terms of rhombic staircase tableaux download download high res image 75kb download download full size image

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

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

shock profiles for the asymmetric simple exclusion process in one dimension the asymmetric simple exclusion process asep on a one dimensional lattice is a system of particles which jump at rates p and 1 p here p 1 2 to adjacent empty sites on their right and left respectively the system is described on suitable macroscopic spatial and temporal scales by the inviscid burgers equation the latter has shock solutions with a discontinuous jump from left density rho to right density rho rho sqrt rho 1 rho rho 1 rho for a special value of the asymmetry given by p 1 p rho 1 rho rho 1 rho the measure is bernoulli with density rho on the left and rho on the right in the weakly asymmetric limit 2p 1 to0 the microscopic width of the shock diverges as 2p 1 1 the stationary measure is then essentially a superposition of bernoulli measures corresponding to a convolution of a density profile described by the viscous burgers equation with a well defined distribution for the location of the second class particle