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

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

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

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

dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition the one dimensional totally asymmetric simple exclusion process tasep is considered we study the time evolution property of a tagged particle in the tasep with the step initial condition calculated is the multi time joint distribution function of its position using the relation of the dynamics of the tasep to the schur process we show that the function is represented as the fredholm determinant we also study the scaling limit the universality of the largest eigenvalue in the random matrix theory is realized in the limit when the hopping rates of all particles are the same it is found that the joint distribution function converges to that of the airy process after the time at which the particle begins to move on the other hand when there are several particles with small hopping rate in front of a tagged particle the limiting process changes at a certain time from the airy process to the process of the largest eigenvalue in the hermitian multi matrix model with external sources

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

a quantization property for static ginzburg landau vortices for any critical point of the complex ginzburg landau functional in dimension 3 we prove that for large coupling constants if the energy of this critical point on a ball of a given radius r is relatively small compared to r log then the ball of half radius contains no vortex the modulus of the solution is larger than we then show how this property can be applied to describe limiting vortices as e 0 2001 john wiley sons inc

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

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

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

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

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

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

convex functionals and the stratification of the singular set of their stationary points abstract we prove partial regularity of stationary solutions and minimizers u from a set r n to a riemannian manifold n for the functional f x u u 2 d x the integrand f is convex and satisfies some ellipticity and boundedness assumptions we also develop a new monotonicity formula and an regularity theorem for such stationary solutions with no restriction on their images we then use the idea of quantitative stratification to show that the k th strata of the singular set of such solutions are k rectifiable

current distribution and random matrix ensembles for an integrable asymmetric fragmentation process we calculate the time evolution of a discrete time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size dependent rates in the continuous time limit the process turns into the totally asymmetric simple exclusion process only pieces of size 1 break off a given cluster we express the exact solution of the master equation for the process in terms of a determinant which can be derived using the bethe ansatz from this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the gaussian unitary ensemble of random matrices this result confirms universality of the scaling form of the current distribution in the kpz universality class and suggests that there is a link between integrable particle systems and random matrix ensembles

lamb s problem at its simplest this article revisits the classical problem of horizontal and vertical point loads suddenly applied onto the surface of a homogeneous elastic half space and provides a complete set of exact explicit formulae which are cast in the most compact format and with the simplest possible structure the formulae given are valid for the full range of poisson9s ratios from 0 to 0 5 and they treat real and complex poles alike as a result of which a single set of formulae suffices and also exact formulae for dipoles can be given

long time asymptotics of the totally asymmetric simple exclusion process we study the long time asymptotics of the relaxation dynamics of the totally asymmetric simple exclusion process on a ring evaluating the asymptotic amplitudes of the local currents by the algebraic bethe ansatz method we find that the relaxation times starting from the step and alternating initial conditions are governed by different eigenvalues of the markov matrix in both cases the scaling exponents of the leading asymptotic amplitudes with respect to the total number of sites are found to be 1 we also study the asymptotics of correlation functions such as the emptiness formation probability

analytic resolution of time domain half space green s functions for internal loads by a displacement potential laplace hankel cagniard transform method a refined yet compact analytical formulation is presented for the time domain elastodynamic response of a three dimensional half space subject to an arbitrary internal or surface force distribution

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

liquid crystals and the monotonicity formula abstract we give numerical evidence that the monotonicity formula holds for the liquid crystal functional and study part of the associated bifurcation diagram

a monotonicity formula and a liouville type theorem for a fourth order supercritical problem we consider liouville type and partial regularity results for the nonlinear fourth order problem

a regularity theory of biharmonic maps in this article we prove the regularity of weakly biharmonic maps of domains in euclidean four space into spheres as well as the corresponding partial regu larity result of stationary biharmonic maps of higher dimensional domains into spheres c 1999 john wiley sons inc in this article we consider for simplicity the class of bihar monic maps from eu clidean domains to spheres we realize the standard spheres s k as unit vectors in r k 1 and consider maps u s k as vector valued functions that are contained in s k the energy functional for biharmonic maps is then j uj 2 dx a locally de fined biharmonic map is a map that is critical with respect to c ompactly supported variations we note that in the case where the domain has dimension four this en ergy functional is conformally invariant and hence conformal maps of euclidean four space are biharmonic in this sense we remark that this definition of bihar monic map depends on the embedding of the target space in euclidean space we

new monotonicity formulas for ricci curvature and applications i we prove three new monotonicity formulas for manifolds with a lower ricci curvature bound and show that they are connected to rate of convergence to tangent cones in fact we show that the derivative of each of these three monotone quantities is bounded from below in terms of the gromov hausdorff distance to the nearest cone the monotonicity formulas are related to the classical bishop gromov volume comparison theorem and perelman s celebrated monotonicity formula for the ricci flow we will explain the connection between all of these moreover we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the green function that we prove this is parallel to the fact that perelman s monotonicity is closely related to the sharp gradient estimate for the heat kernel of li yau in cm4 one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross sections of einstein manifolds finally there are obvious parallelisms between our monotonicity and the positive mass theorem of schoen yau and witten

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

an elementary partial regularity proof for vector valued obstacle problems soit r n et m r n des ensembles ouverts bornes on suppose n 3 n 2 et on requiert que la frontiere de m est une hyper surface lisse de classe c 2 l espace de fonction h 1 2 m des fonctions de comparaison consiste en toutes les fonctions de sobolev v h 1 2 r n avec la propriete v x m pour presque tous les points x dans si u h 1 2 m minimise localement l energie alors il y a un sous ensemble ouvert 0 de tel que u c 1 0 r n pour tout 0 1 h n 2 0 0 et x 0 si et seulement si lim 1 0 r 2 n br x du 2 dz 0

lamb s problem on random mass density fields with fractal and hurst effects this paper reports on a generalization of lamb s problem to a linear elastic infinite half space with random fields rfs of mass density subject to a normal line load both uncorrelated and correlated with fractal and hurst characteristics rfs without any weak noise restrictions are proposed cellular automata ca is used to simulate the wave propagation ca is a local computational method which for rectangular discretization of spatial domain is equivalent to applying the finite difference method to the governing equations of classical elasticity we first evaluate the response of ca to an uncorrelated mass density field more commonly known as white noise of varying coarseness as compared to ca s node density we then evaluate the response of ca to multiscale mass density rfs of cauchy and dagum type these fields are unique in that they are able to model and decouple the field s fractal dimension and hurst parameter we focus on stochastic imperfection sensitivity we determine to what extent the fractal or the hurst parameter is a significant factor in altering the solution to the planar stochastic lamb s problem by evaluating the coefficient of variation of the response when compared with the coefficient of variation of the rf

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

the stress energy tensor for polyharmonic maps abstract we derive the stress energy tensor for polyharmonic maps between riemannian manifolds moreover we employ the stress energy tensor to characterize polyharmonic maps where we pay special attention to triharmonic maps

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

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

bethe ansatz calculation of the spectral gap of the asymmetric exclusion process we present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half filled ring of size l by using the bethe ansatz we show that in the large l limit the bethe equations reduce to a simple transcendental equation involving the polylogarithm a classical special function by solving that equation the gap and the dynamical exponent are readily obtained our method can be extended to a system with an arbitrary density of particles

total current fluctuations in the asymmetric simple exclusion process a limit theorem for the total current in the asymmetric simple exclusion process asep with step initial condition is proven this extends the result of johansson on tasep to asep

vertex models tasep and grothendieck polynomials we examine the wavefunctions and their scalar products of a one parameter family of integrable five vertex models at a special point of the parameter the model investigated is related to an irreversible interacting stochastic particle system the so called totally asymmetric simple exclusion process tasep by combining the quantum inverse scattering method with a matrix product representation of the wavefunctions the on off shell wavefunctions of the five vertex models are represented as a certain determinant form up to some normalization factors we find that the wavefunctions are given by grothendieck polynomials which are a one parameter deformation of schur polynomials introducing a dual version of the grothendieck polynomials and utilizing the determinant representation for the scalar products of the wavefunctions we derive a generalized cauchy identity satisfied by the grothendieck polynomials and their duals several representation theoretical formulae for the grothendieck polynomials are also presented as a byproduct the relaxation dynamics such as green functions for the periodic tasep are found to be described in terms of the grothendieck polynomials

exact relaxation dynamics in the totally asymmetric simple exclusion process the relaxation dynamics of the one dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition analyzing the time evolution of the local particle densities and currents by the bethe ansatz method we examine their full relaxation dynamics as a result we observe peculiar behaviors such as the emergence of a ripple in the density profile and the existence of the excessive particle currents moreover by making a finite size scaling analysis of the asymptotic amplitudes of the local densities and currents we find the scaling exponents with respect to the total number of sites to be ensuremath 3 2 and ensuremath 1 respectively

harmonic maps from noncompact riemannian manifolds with non negative ricci curvature outside a compact set in this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete noncompact riemannian manifold m g with sobolev constant s2 m 0 and ricci curvature ric m 0 outside some compact subset into a complete manifold of nonpositive curvature or a regular ball in particular we prove the uniqueness and existence of bounded harmonic functions on m g

the uniqueness of tangent cones for yang mills connections with isolated singularities we proved a uniqueness theorem of tangent connections for a yang mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity we also obtained control over the rate of the asymptotic convergence of the connection to the tangent connection if furthermore the connection is stationary or the tangent connection is integrable with a stronger result in the latter case there are parallel results for the cones at infinity of a yang mills connection on an asymptotically flat manifold we also gave an application of our methods to the yang mills flow and proved that the yang mills flow exists for all time and has asymptotic limit if the initial value is close to a smooth local minimizer of the yang mills functional

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

a monotonicity formula for stationary biharmonic maps we give a rigorous proof of the monotonicity formula of s y a chang l wang and p yang 3 for extrinsically stationary biharmonic maps of class w2 2

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

biaxial escape in nematics at low temperature abstract in the present work we study minimizers of the landau de gennes free energy in a bounded domain r 3 we prove that at low temperature minimizers do not vanish even for topologically non trivial boundary conditions this is in contrast with a simplified ginzburg landau model for superconductivity studied by bethuel brezis and helein merging this with an observation of canevari we obtain as a corollary the occurrence of biaxial escape the tensorial order parameter must become strongly biaxial at some point in in particular while it is known that minimizers cannot be purely uniaxial we prove the much stronger and physically relevant fact that they lie in a different homotopy class

compactness theorems for coupled yang mills fields in this paper we consider coupled yang mills fields on vector bundle e over compact riemannian manifold m under appropriate conditions on the curvature and the higgs field two compactness theorems are proved

monotonicity formulas for extrinsic triharmonic maps and the triharmonic lane emden equation abstract we derive monotonicity formulas and e regularity results for extrinsic triharmonic maps and triharmonic lane emden equations as an application we prove partial regularity results for both types of equations in the supercritical regime

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process