improved bounds for the transition temperature of directed polymers in a finite dimensional random medium we consider the problem of directed polymers in a random medium of a finitedimensional lattice in the high temperature phase of this system it is known that the annealed and quenched free energies coincide upper bounds on the transition temperature to a low temperature phase had previously been obtained by calculating the first two moments z and z2 of the partition function we improve these bounds by estimating noninteger moments z for 1 2

solving singularly perturbed boundary value problems by spline in tension the difference scheme via spline in tension for the problem y p x y f x p x s 0 y 0 0 y 1 1 is derived the error of the form o h min h is obtained when p x p const the corresponding spline in tension achieves the second order of the global uniform convergence

a weak galerkin finite element method with polynomial reduction the weak galerkin wg is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions in the implementation the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom the accuracy and the computational complexity of the corresponding wg scheme is significantly impacted by the selection of such polynomials this paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation for illustrative purpose the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction consequently a new weak galerkin finite element method is proposed and analyzed error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper presents some numerical results to demonstrate the power of the wg method in dealing with finite element partitions with arbitrary polygons in 2d or polyhedra in 3d the numerical examples include various finite element partitions such as triangular mesh quadrilateral mesh honeycomb mesh in 2d and mesh with deformed cubes in 3d

a robust wg finite element method for convection diffusion reaction equations this paper proposes and analyzes a weak galerkin wg finite element method for 2 and 3 dimensional convection diffusion reaction problems on conforming or nonconforming polygon polyhedral meshes the wg method uses piecewise polynomial approximations of degrees k k 0 for both the scalar function and its trace on the inter element boundaries we show that the method is robust in the sense that the derived a priori error estimates is uniform with respect to the coefficients for sufficient smooth true solutions numerical experiments confirm the theoretical results

an almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction diffusion problem this paper is concerned with a high order convergent discretization for the semilinear reaction diffusion problem varepsilon 2 u b x u 0 for x in 0 1 subject to u 0 u 1 0 where varepsilon in 0 1 we assume that b u x u b 0 2 0 on 0 1 times bbb r 1 which guarantees uniqueness of a solution to the problem asymptotic properties of this solution are discussed we consider a polynomial based three point difference scheme on a simple piecewise equidistant mesh of shishkin type existence and local uniqueness of a solution to the scheme are analysed we prove that the scheme is almost fourth order accurate in the discrete maximum norm uniformly in the perturbation parameter varepsilon we present numerical results in support of this result

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

fluctuations of extensive functions of quenched random couplings an extensive quantity is a family of functions v of random parameters indexed by the finite regionsv subsets of d over which v are additive up to corrections satisfying the boundary estimate stated below it is shown that unless the randomness is nonessential in the sense that lim v v has a unique value in the absolute i e not just probabilistic sense the variance of such a quantity grows as the volume ofv of particular interest is the free energy of a system with random couplings for such v bounds are derived also for the generating functione et in a separate application variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment

weak galerkin method for the biot s consolidation model abstract we develop a weak galerkin wg finite element method for the biot s consolidation model in the classical displacement pressure two field formulation weak galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations backward euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme we study the well posedness of the linear system at each time step and also derive the overall optimal order convergence of the wg formulation such wg scheme is designed on general shape regular polytopal meshes and provides stable and oscillation free approximation for the pressure without special treatment numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak galerkin finite element method

superdiffusivity for a brownian polymer in a continuous gaussian environment this paper provides information about the asymptotic behavior of a one dimensional brownian polymer in random medium represented by a gaussian field w on r r assumed to be white noise in time and function valued in space according to the behavior of the spatial covariance of w we give a lower bound on the power growth wandering exponent of the polymer when the time parameter goes to infinity the polymer is proved to be superdiffusive with a wandering exponent exceeding any 3 5

directed polymers in a random medium 1 d expansion and the n tree approximation the authors develop a method which allows one to obtain 1 d expansions for the problem of directed polymers in a random medium using this method they expand thermal properties such as the free energy the specific heat the overlaps and the fluctuations of the transverse displacement their results are consistent with the existence of a finite upper critical dimension above which the low temperature phase is mean field like i e characterised by a broad distribution of overlaps broken replica symmetry and transverse fluctuations which scale as the length of the polymer

on the partition function of a directed polymer in a gaussian random environment the purpose of this work is the study of the partition function of a dimensional lattice directed polymer in a gaussian random environment being the inverse of temperature in the low dimensional cases we prove that for all the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0 in the high dimensional case a lower tail of has been obtained for small furthermore we express some thermodynamic quantities in terms of the path measure alone

the largest liapunov exponent for random matrices and directed polymers in a random environment we study the largest liapunov exponent for products of random matrices the two classes of matrices considered are discrete d dimensional laplacians with random entries and symplectic matrices that arise in the study ofd dimensional lattices of coupled nonlinear oscillators we derive bounds on this exponent for all dimensions d and we show that ifd 3 and the randomness is not too strong one can obtain an explicit formula for the largest exponent in the thermodynamic limit our method is based on an equivalence between this problem and the problem of directed polymers in a random environment

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

fluctuations of the solutions to the kpz equation in dimensions three and higher we prove using probabilistic techniques and analysis on the wiener space that the large scale fluctuations of the kpz equation in d ge 3 with a small coupling constant driven by a white in time and colored in space noise are given by the edwards wilkinson model this gives an alternative proof that avoids perturbation expansions to the results of magnen and unterberger j stat phys 171 543 598 2018

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

directed polymers on trees a martingale approach the authors use martingale methods and simple convexity arguments to compute rigorously the limiting free energy in the problem of directed polymers on a tree the limit is a degenerate random variable and convergence holds almost surely the only assumption on the common distribution of the random potentials attached to the bonds of the tree is that its laplace transform exists everywhere in 0 infinity

improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem a class of singularly perturbed two point boundary value problems bvps for second order ordinary differential equations in self adjoint form arising in the study of chemical catalysis and michaelis menten process in biology is considered in order to solve them a numerical method as in appl math comput 55 1993 31 is proposed the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the given equations over these regions separately as three two point bvps numerically the inner region problems are solved using a fitted operator method uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 whereas the outer region problem is solved using a classical central difference method the boundary conditions at the transition points are obtained using the zero order asymptotic expansion approximation to the solution of the problem this method is well suited for parallel computing and an algorithm for the same is given error estimates are derived for the numerical solution some schemes for self adjoint equations in conservation form are given using newton s method of quasilinearization a class of semilinear problems are also solved numerical experiments are conducted it is found that the present method performs better than the fitted mesh method fitted numerical methods for singular perturbation problems world scientific singapore 1996 and higher order method numeriche mathematik 56 1990 675

a note on the diffusion of directed polymers in a random environment a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford 3 and if the disorder is small enough this simplifies and extends an approach by j imbrie and t spencer

directed polymers in a random environment with heavy tails we study the model of directed polymers in a random environment in 1 1 dimensions where the distribution at a site has a tail that decays regularly polynomially with power where 0 2 after proper scaling of temperature 1 we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced we prove that this region has a weak limit under linear scaling and identify the limiting distribution as an indexed family of measures on lipschitz curves lying inside the 45 rotated square with unit diagonal in particular this shows order n transversal fluctuations of the polymer if and only if is small enough we find that there exists a random critical temperature below which but not above which the effect of the environment is macroscopic the results carry over to d 1 dimensions for d 1 with minor modifications 2010 wiley periodicals inc

weak galerkin finite element method for biots consolidation problem in this paper a fully discrete weak galerkin wg finite element method is proposed to solve biots consolidation problem where weakly defined gradient and divergence operators over discontinuous functions are introduced plpl l1 finite element combination is used for the displacement and pressure approximations in the interior of the elements and plpl1 combination for the corresponding trace approximations on the interfaces of the finite element partition the existence and uniqueness of the discrete linear system at each time step is derived and error estimates for the approximation of displacement and pressure are obtained numerical experiments confirm the theoretical results and show that the proposed wg method is capable of overcoming pressure oscillations

almost sure central limit theorem for directed polymers and random corrections we consider a general model of directed polymers on the lattice weakly coupled to a random environment we prove that the central limit theorem holds almost surely for the discrete time random walk x t associated to the polymer moreover we show that the random corrections to the cumulants of x t are finite starting from some dimension depending on the index of the cumulants and that there are corresponding random corrections of order in the asymptotic expansion of the expectations of smooth functions of x t full proofs are carried out for the first two cumulants we finally prove a kind of local theorem showing that the ratio of the probabilities of the events to the corresponding probabilities with no randomness in the region of moderate deviations from the average drift bt are for almost all choices of the environment uniformly close as to a functional of the environment as seen from t y

rate of convergence for polymers in a weak disorder we consider directed polymers in random environment on the lattice z d at small inverse temperature and dimension d 3 then the normalized partition function w n is a regular martingale with limit w we prove that n d 2 4 w n w w n converges in distribution to a gaussian law both the polynomial rate of convergence and the scaling with the martingale w n are different from those for polymers on trees

localization in log gamma polymers with boundaries consider the directed polymer in one space dimension in log gamma environment with boundary conditions introduced by seppalainen ann probab 40 1 19 73 2012 in the equilibrium case we prove that the end point of the polymer converges in law as the length increases to a density proportional to the exponent of a zero mean random walk this holds without space normalization and the mass concentrates in a neighborhood of the minimum of this random walk we have analogous results out of equilibrium as well as for the middle point of the polymer with both ends fixed the existence and the identification of the limit relies on the analysis of a random walk seen from its infimum

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

large deviations for brownian motion in a random scenery we prove large deviations principles in large time for the brownian occupation time in random scenery frac 1 t int 0 t xi b s ds the random field is constant on the elements of a partition of d into unit cubes these random constants say left lbrace xi j j in mathbb z d right rbrace consist of i i d bounded variables independent of the brownian motion bs s 0 this model is a time continuous version of kesten and spitzer s random walk in random scenery we prove large deviations principles in quenched and annealed settings

directed last passage percolation with discontinuous weights we prove that a directed last passage percolation model with discontinuous macroscopic non random inhomogeneities has a continuum limit that corresponds to solving a hamilton jacobi equation in the viscosity sense this hamilton jacobi equation is closely related to the conservation law for the hydrodynamic limit of the totally asymmetric simple exclusion process we also prove convergence of a numerical scheme for the hamilton jacobi equation and present an algorithm based on dynamic programming for finding the asymptotic shapes of maximal directed paths

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection diffusion problems on shishkin mesh this article studies the numerical solution of singularly perturbed delay parabolic convection diffusion initial boundary value problems since the solution of these problems exhibit regular boundary layers in the spatial variable we use the piecewise uniform shishkin mesh for the discretization of the domain in the spatial direction and uniform mesh in the temporal direction the time derivative is discretized by the implicit euler scheme and the spatial derivatives are discretized by the hybrid scheme for the proposed scheme the stability analysis is carried out and parameter uniform error estimates are derived numerical examples are presented to show the accuracy and efficiency of the proposed scheme

directed polymers in random environment are diffusive at weak disorder in this paper we consider directed polymers in random environment with discrete space and time for transverse dimension at least equal to 3 we prove that diffusivity holds for the path in the full weak disorder region i e where the partition function differs from its annealed value only by a non vanishing factor deep inside this region we also show that the quenched averaged energy has fluctuations of order 1 in complete generality arbitrary dimension and temperature we prove monotonicity of the phase diagram in the temperature

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

surface structure of random aggregates on the cayley tree growth through ballistic aggregation and biased diffusion limited aggregation is investigated on the cayley tree for a general branching ratio it is shown that the surface width of a ballistic aggregate remains finite of order one as the cluster mass goes to infinity dla with an attractive bias is treated approximately for non zero bias strength the surface width is asymptotically finite in the limit of isotropic diffusion zero bias a roughening transition occurs which can be described in terms of a single diverging length scale

strong localization and macroscopic atoms for directed polymers in this article we derive strong localization results for directed polymers in random environment we show that at low temperature the polymer measure is asymptotically concentrated at a few points of macroscopic mass we call these points epsilon atoms these results are derived assuming weak conditions on the tail decay of the random environment

a remark on the bound for the free energy of directed polymers in random environment in 1 2 dimension we consider the behavior of the quantity p the free energy of directed polymers in random environment in 1 2 dimension where is inverse temperature we know that the free energy is strictly negative when is not zero in this paper we will prove that p is bounded from above by exp c 2 for small where ce 0 is a constant depending on e 0 also we will suggest a strategy to get a sharper asymptotics we denote by f p a probability space

a computational method for self adjoint singular perturbation problems using quintic spline singularly perturbed self adjoint boundary value problems are considered in this article a difference scheme based on quintic spline is proposed this scheme is applied to the subproblems obtained from the given problem by dividing the whole domain into nonoverlapping subdomains the proposed scheme is of fourth order convergent and more suitable for parallel computers stability and convergence of the method are discussed numerical examples are provided to show the efficiency and accuracy

probability distribution of the free energy of a directed polymer in a random medium we calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder by using the fact that the nth moment of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length l we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n for n small and n approximately lgamma 1 2 the moments take a scaling form which allows us to describe all the fluctuations of order 1 l of the free energy per unit length of the directed polymer the distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process indicating that it is characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimensions

finite size effects in random energy models and in the problem of polymers in a random medium by the use of traveling wave equations we calculate the finite size corrections to the free energy of random energy models in their low temperature phases and in the neighborhood of the transition temperature we find that although the extensive part of the free energy does not show any critical behavior when the temperature approaches its transition value the finite size corrections signal the transition by becoming singular we obtain a scaling form for these finite size corrections valid in the limitn andt t c by considering a generalized random energy model in the limit of a very large number of steps we obtain results for the finite size corrections in the problem of a polymer in a random medium

weak galerkin method with implicit θ schemes for second order parabolic problems abstract we introduce a new weak galerkin finite element method whose weak functions on interior edges are double valued and approximation spaces are based on pk t pk e rtk t elements it is natural to develop a semi discrete stable scheme for parabolic problems and then fully discrete approaches are formulated with implicit schemes in time for 1 2 1 which include first order backward euler 1 and second order crank nicolson schemes 1 2 furthermore optimal convergence rates in the l2 and energy norms are derived numerical results are given to verify the theory

the edwards wilkinson limit of the random heat equation in dimensions three and higher we consider the heat equation with a multiplicative gaussian potential in dimensions d 3 we show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity we also prove that the renormalized large scale random fluctuations are described by the edwards wilkinson model that is the stochastic heat equation she with additive white noise with an effective variance

mean field theory of directed polymers with random complex weights we show that for the problem of directed polymers on a tree with i i d random complex weights on each bond three possible phases can exist the phase of a particular system is determined by the distribution of the random weights for each of these three phases we give the expression of the free energy per unit length in the limit of infinitely long polymers our proofs require several hypotheses on the distribution most importantly that the amplitude and the phase of each complex weight be statistically independent the main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy we illustrate our results by some examples and discuss possible generalizations to a larger class of distributions to random energy models and to the finite dimensional case we note that our results are not in agreement with the predictions of a recent replica approach to a similar problem

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

infinite moments of the partition function for random walks in a random potential we consider a system of random walks directed polymers in a potential which is random in space and time investigations into untypical behavior and the inverse of the partition function leads to the result that regardless of the strength of the potential and in all dimensions all higher moments of the partition function are unbounded as time goes to infinity one can interpret this result in a many body quantum mechanical setting as saying that no matter how small the strength of an attractive potential in any dimension one can find n the number of particles large enough so that there is a bound state

quenched free energy and large deviations for random walks in random potentials we study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps the potential can be unbounded and can depend on a few steps of the walk directed undirected and stretched polymers as well as random walk in random environment are covered the restriction needed is on the moment of the potential in relation to the degree of mixing of the ergodic environment we derive two variational formulas for the limiting quenched free energy and prove a process level quenched large deviation principle ldp for the empirical measure as a corollary we obtain ldps for types of random walks in random environments not covered by earlier results 2012 wiley periodicals inc

diffusion of directed polymers in a strong random environment we consider a system of random walks or directed polymers interacting with an environment which is random in space and time it was shown by imbrie and spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the bernoulli distribution under the same assumption on the random environment as that of lmbrie and spencer we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment more generally we can prove that if the random environment is bounded and if the supremum of the support of the distribution has a positive mass then there is an integer do such that in dimensions higher than do the behavior of the random polymer is always diffusive

the number of open paths in an oriented ρ percolation model we study the asymptotic properties of the number of open paths of length n in an oriented percolation model we show that this number is en 1 o 1 as n the exponent is deterministic it can be expressed in terms of the free energy of a polymer model and it can be explicitly computed in some range of the parameters moreover in a restricted range of the parameters we even show that the number of such paths is n 1 2wen 1 o 1 for some nondegenerate random variable w we build on connections with the model of directed polymers in random environment and we use techniques and results developed in this context

on the brownian directed polymer in a gaussian random environment in this paper we introduce a model of brownian polymer in a continuous random environment the asymptotic behavior of the partition function associated to this polymer measure is studied and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment some further developments concerning some concentration inequalities for the partition function are given for the weak disorder regime

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

directed paths on percolation clusters we consider a variant of the problem of directed polymers on a disordered lattice in which the disorder is geometrical in nature in particular we allow a finite probability for each bond to be absent from the lattice we show through the use of numerical and scaling arguments on both euclidean and hierarchical lattices that the model has two distinct scaling behaviors depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold we are particularly interested in the exponents and defined by f t and x t describing the free energy and transverse fluctuations respectively above the percolation threshold the scaling behavior is governed by the standard random energy exponents 1 3 and 2 3 in 1 1 dimensions at the percolation threshold we predict and verify numerically in 1 1 dimensions the exponents 1 2 and v v where v and v are the directed percolation exponents in addition we predict the absence of a free phase in any dimension at the percolation threshold

a brownian motion version of the directed polymer problem consider a brownian particle in three dimensions in a random environment the environment is determined by a potential random in space and time it is shown that at small noise the large time behavior of the particle is diffusive the diffusion constant depends on the environment this work generalizes previous results for random walk in a random environment in these results the diffusion constant does not depend on the environment

the burgers equation with a random force and a general model for directed polymers in random environments the study of the burgers equation with a random force leads via a hopf cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential the latter can be studied by means of a general model of directed polymers in random environments with two point random potentials these models exhibit a gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and burgers equations

the two dimensional stochastic heat equation renormalizing a multiplicative noise we study in two space dimensions the heat equation with a random potential that is a white noise in space and time we introduce a regularization of the noise and prove that by a suitable renormalization of the coupling coefficient the covariance has a non trivial limit when the regularization is removed the limit is described in terms of a two body schrodinger operator with singular interaction

localization transition for polymers in poissonian medium we study a model of directed polymers in random environment in dimension 1 d given by a brownian motion in a poissonian potential we study the effect of the density and the strength of inhomogeneities respectively the intensity parameter of the poisson field and the temperature inverse our results are i fine information on the phase diagram with quantitative estimates on the critical curve ii pathwise localization at low temperature and or large density iii complete localization in a favourite corridor for large 2 and bounded

directed polymers in a random environment some results on fluctuations we consider a polymer model on where to each edgee is associated a random variable v e a polymer configuration is represented by a directed pathr and has a weight exp e r e with 1 t the inverse temperature we extend some rigorous results that have been obtained for the ground state of this model to finite temperatures in particular we obtain some upper and lower bounds on sample to sample free energy fluctuations and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations

brownian directed polymers in random environment we study the thermodynamics of a continuous model of directed polymers in random environment the environment is given by a space time poisson point process whereas the polymer is defined in terms of the brownian motion we mainly discuss i the normalized partition function its positivity in the limit which characterizes the phase diagram of the model ii the existence of quenched lyapunov exponent its positivity and its agreement with the annealed lyapunov exponent iii the longitudinal fluctuation of the free energy some of its relations with the overlap between replicas and with the transversal fluctuation of the path the model considered here enables us to use stochastic calculus with respect to both brownian motion and poisson process leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram we also relate our model to some formulation of the kardar parisi zhang equation more precisely the stochastic heat equation our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super diffusivity in dimension one we also obtain an almost sure large deviation principle for the polymer measure

almost sure central limit theorem for a markov model of random walk in dynamical random environment we consider a model of random walk on 2 in a dynamical random environment described by a field t x t x 1 the random walk transition probabilities are taken as p x t 1 y x t x t p 0 y x c y x x we assume that the variables t x t x 1 are i i d that both p 0 u and c u s are finite range in u and that the random term c u is small and with zero average we prove that the c l t holds almost surely with the same parameters as for p 0 for all 2 for 3 there is a finite random i e dependent on correction to the average of x t and there is a corresponding random correction of order to the c l t for 5 there is a finite random correction to the covariance matrix of x t and a corresponding correction of order to the c l t proofs are based on some new l p estimates for a class of functionals of the field

a robust computational method for singularly perturbed coupled system of reaction diffusion boundary value problems abstract in this paper we devise a robust computational method for singularly perturbed system of reaction diffusion boundary value problems bvps we use cubic spline on nonuniform mesh to obtain the difference scheme and apply it to the system of bvps on a piecewise uniform shishkin mesh on the whole domain we observed that the cubic spline scheme is not stable in the regular outer region where the mesh is coarse to overcome this difficulty we use the classical central difference scheme only for that portion and the cubic spline scheme elsewhere this newly proposed scheme is uniformly stable throughout the domain and provides second order uniform convergence results we derive the uniform error estimate for this scheme and apply it to a test problem to verify the efficiency and accuracy of the method

random walks in quenched i i d space time random environment are always a s diffusive we consider a general model of discrete time random walk x t on the lattice 1 in a random environment t x t x 1 with i i d components t x previous results on the a s validity of the central limit theorem for the quenched model required a small stochasticity condition in this paper we show that the result holds provided only that an obvious non degeneracy condition is met the proof is based on the analysis of a suitable generating function which allows to estimate l 2 norms by contour integrals

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

a remark on diffusion of directed polymers in random environments we consider a system of random walks or directed polymers interacting with an environment which is random in space and time under minimal assumptions on the distribution of the environment we prove that this system has diffusive behavior with probability one if d 2 and fl flo where flo is defined in terms of the probability that the symmetric nearest neighbor random walk on the d dimensional integer lattice ever returns to its starting point we also obtain a precise estimate for the mean square displacement of this system

anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media the central limit theorem for a model of discrete time random walks on the lattice in a fluctuating random environment was proved for almost all realizations of the space time nvironment for all 1 in bmp1 and for all 1 in bbmp in bmp1 it was proved that the random correction to the average of the random walk for 3 is finite in the present paper we consider the cases 1 2 and prove the central limit theorem as t for the random correction to the first two cumulants the rescaling factor for theaverage is for 1 and ln t for 2 for the covariance it is 1 2

schramm loewner evolution theory of the asymptotic behaviors of 2 1 dimensional das sarma tamborenea model abstract a mass of theoretical analysis and numerical calculation on 2 1 dimensional das sarma tamborenea model have been carried out over the years a problem about asymptotical behaviors that this model belongs to which universality classes is still interesting and controversial the schramm loewner evolution theory is an innovative point of view to describe the conformal invariance of contour lines of saturated rough surfaces in this paper we used sle theory to analyze the conformal invariance of the 2 1 dimensional dt model a noise reduction technique was also utilized to get better scaling behavior the results show that the contour lines of 2 1 dimensional dt model satisfy the property of conformal invariance and belongs to 4 universality class the relation d f 1 8 is also satisfied for the fractal dimension and diffusion coefficient of the contour lines

a weak galerkin finite element method for singularly perturbed convection diffusion reaction problems in this article a new weak galerkin finite element method is introduced to solve convection diffusion reaction equations in the convection dominated regime our method is highly flexible by allow

on the perturbation expansion of the kpz equation we present a simple argument to show that the function of the d dimensional kpz equation d 2 is to all orders in perturbation theory given by beta g r d 2 g r 2 8 pi d 2 text gamma 2 d 2 g r 2 neither the dynamical exponent z nor the roughness exponent have any correction in any order of perturbation theory this shows that standard perturbation theory cannot attain the strong coupling regime and in addition breaks down at d 4 we also calculate a class of correlation functions exactly

growth anomaly and its implications abstract the growth mechanism of interfaces in nature may be anomalous in the sense that the interface s roughness is not a universal scaling function this so called growth anomaly can be traced to the non existence of a continuum limit of the growth theory various aspects of the anomaly are discussed and some experimental verifications are suggested

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations a method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order o d e s an improvement by a factor of n 1 can be obtained where is the small parameter andn is the order of the asymptotic approximation for a small amount of extra work numerical experiments are presented

a kpz cocktail shaken not stirred the stochastic partial differential equation proposed nearly three decades ago by kardar parisi and zhang kpz continues to inspire intrigue and confound its many admirers here we i pay debts to heroic predecessors ii highlight additional experimentally relevant aspects of the recently solved 1 1 kpz problem iii use an expanding substrates formalism to gain access to the 3d radial kpz equation and lastly iv examining extremal paths on disordered hierarchical lattices set our gaze upon the fate of d infty kpz clearly there remains ample unexplored territory within the realm of kpz and for the hearty much work to be done especially in higher dimensions where numerical and renormalization group methods are providing a deeper understanding of this iconic equation

large deviation rate functions for the partition function in a log gamma distributed random potential we study right tail large deviations of the logarithm of the partition function for directed lattice paths in i i d random potentials the main purpose is the derivation of explicit formulas for the 1 1 dimensional exactly solvable case with log gamma distributed random weights along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions

scaling for a one dimensional directed polymer with boundary conditions we study a 1 1 dimensional directed polymer in a random environment on the integer lattice with log gamma distributed weights among directed polymers this model is special in the same way as the last passage percolation model with exponential or geometric weights is special among growth models namely both permit explicit calculations with appropriate boundary conditions the polymer with log gamma weights satisfies an analogue of burke s theorem for queues building on this we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path in the case where the boundary conditions are present and both endpoints of the polymer path are fixed for the polymer without boundary conditions and with either fixed or free endpoint we get the expected upper bounds on the exponents

improvement of numerical solution of selfadjoint singular perturbation problems by incorporation of asymptotic approximations a numerical method for singularly perturbed two point boundary value problems for second order ordinary differential equations without a first derivative term arising in the study of chemical catalysis and michaelis menten process in biology is proposed in this method booster method an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution uniform error estimates are derived for this method when implemented in known difference schemes an improvement by a factor of n 1 can be obtained where is the small parameter and n is the order of the asymptotic approximation for a small amount of extra work numerical results are presented in support of this claim

critical discussion of the two loop calculations for the kardar parisi zhang equation in this article we perform a careful analysis of the renormalization procedure used in existing calculations to derive critical exponents for the kpz equation at 2 loop order this analysis explains the discrepancies between the results of the different groups the correct critical exponents in d 2 epsilon dimensions at the crossover between weak and strong coupling regime are zeta o epsilon 3 and z 2 o epsilon 3 no strong coupling fixed point exists at 2 loop order

nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices we prove the moments of the directed polymer partition function gz using an exact position space renormalization group scheme on a hierarchical lattice after sufficient iteration the characteristic functionf n ln gzn of the probability z converges to a stable limitf n for smalln the limiting behavior is independent of the initial distribution while for largen f n is completely determined by it and is thus nonuniversal there is a smooth crossover between the two regimes for small effective dimensions and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution for large effective dimensions there is a sharp transition between the two regimes and analytical continuation from integer moments ton 0 is not possible replica arguments can account for most features of the observed results