bases of a free semimodule are small abstract a generating set g of a left semimodule s over a semiring r is called a basis if no proper subset of g generates s we prove that r n has no basis of cardinality exceeding qn where q is the largest cardinality of bases of r

exponential decay toward equilibrium via entropy methods for reaction diffusion equations abstract in this work we show how the entropy method enables to get in an elementary way and without linearization estimates of exponential decay towards equilibrium for solutions of reaction diffusion equations corresponding to a reversible reaction explicit rates of convergence combining the dissipative effects of diffusion and reaction are given

kardar parisi zhang universality class and the anchored toom interface we revisit the anchored toom interface and use kardar parisi zhang scaling theory to argue that the interface fluctuations are governed by the airy1 process with the role of space and time interchanged the predictions which contain no free parameter are numerically well confirmed for space time statistics in the stationary state in particular the spatial fluctuations of the interface computed numerically agree well with those given by the goe edge distribution of tracy and widom

bases in semimodules over commutative semirings abstract in this paper the bases of a semimodule over a commutative semiring r are investigated some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over r are given the different possible cardinalities for a basis in a finitely generated free semimodule over r are considered and some equivalent descriptions are obtained for a commutative semiring r satisfying the property that any two bases for a finitely generated free r semimodule have the same cardinality partial results obtained in the paper develop and generalize the corresponding results for commutative join semirings and for commutative zerosumfree semirings

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

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

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

the dirichlet problem of the quantum drift diffusion model abstract we study the quantum drift diffusion model a fourth order parabolic system with dirichlet boundary conditions using a semi discretization approximate method with a compact argument and applying a new entropy estimate we prove the existence of global regular weak solutions

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

time rescaling and asymptotic behavior of some fourth order degenerate diffusion equations in this paper we investigate the large times behavior of weak solutions to the fourth order degenerate parabolic equation ut u nuxxx x modeling the evolution of thin films in particular for all n 0 we prove exponential decay of u x t towards its mean value 1 u dx in l1 norm for long times and we give the explicit n dependent rate of decay the result is based on classical entropy estimates and on detailed lower bounds for the entropy production

the airy1 process is not the limit of the largest eigenvalue in goe matrix diffusion using a systematic approach to evaluate fredholm determinants numerically we provide convincing evidence that the airy1 process arising as a limit law in stochastic surface growth is not the limit law for the evolution of the largest eigenvalue in goe matrix diffusion

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

universal aspects of curved flat and stationary state kardar parisi zhang statistics motivated by the recent exact solution of the stationary state kardar parisi zhang kpz statistics by imamura and sasamoto phys rev lett 108 190603 2012 as well as a precursor experimental signature unearthed by takeuchi phys rev lett 110 210604 2013 we establish here the universality of these phenomena examining scaling behaviors of directed polymers in a random medium the stochastic heat equation with multiplicative noise and kinetically roughened kpz growth models we emphasize the value of cross kpz class universalities revealing crossover effects of experimental relevance finally we illustrate the great utility of kpz scaling theory by an optimized numerical analysis of the ulam problem of random permutations

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

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

rigorous derivation of domain growth kinetics without conservation laws the time evolution of the ising model that describes shrinking domains is studied a singly connected domain of ising spins embedded in a sea of the opposite phase develops att 0 according to a dynamic rule that does not allow its perimeter to increase at long enough times the domain disappears we show that the average lifetime of such a domain is proportional to its area we also consider thet 0 dynamics of a single infinite quadrant the area of the quadrant decreases during the time evolution and we show that the area lost grows linearly with time we solve a first passage time problem as well that is we calculate the average time it takes for the area lost to reach a given value for the first time lastly we map the infinite quadrant model onto a diffusion problem with exclusion in one dimension this latter problem is mapped onto a critical six vertex model

scaling limit for the space time covariance of the stationary totally asymmetric simple exclusion process the totally asymmetric simple exclusion process tasep on the one dimensional lattice with the bernoulli measure as initial conditions 0 1 is stationary in space and time let nt j be the number of particles which have crossed the bond from j to j 1 during the time span 0 t for open image in new window we prove that the fluctuations of nt j for large t are of order t1 3 and we determine the limiting distribution function open image in new window which is a generalization of the gue tracy widom distribution the family open image in new window of distribution functions have been obtained before by baik and rains in the context of the png model with boundary sources which requires the asymptotics of a riemann hilbert problem in our work we arrive at open image in new window through the asymptotics of a fredholm determinant open image in new window is simply related to the scaling function for the space time covariance of the stationary tasep equivalently to the asymptotic transition probability of a single second class particle

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

extending orthogonal subsets of semimodules abstract let r be a commutative semiring and s be an r semimodule with inner product operation a subset b s is called standard orthogonal if for any different b c b b b is a unit and b c 0 tan posed the following problem characterize semirings r such that for every r semimodule with standard orthogonal basis and every standard orthogonal set b s there is a standard orthogonal basis of s containing b in this paper we present a negative result on this problem namely we show that such a description is not possible in terms of first order logic also we provide several interesting examples related to the problem

inner products on semimodules over a commutative semiring abstract in this paper the inner products on the semimodules over a commutative semiring are investigated some characterizations for inner products and for orthogonal sets in the semimodules are given and standard orthogonal basis and orthogonal subsemimodule are discussed in particular an equivalent description is obtained for a semifield s satisfying the property that every standard orthogonal set in a finitely generated semimodule m over s can be extended to a standard orthogonal basis for m also the analogue of the kronecker capelli theorem is proved to be valid for a matrix equation over a commutative semiring partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n dimensional vectors over commutative zerosumfree semirings

superdiffusivity of asymmetric exclusion process in dimensions one and two we prove that the diffusion coefficient for the asymmetric exclusion process diverges at least as fast as t 1 4 in dimension d 1 and logt 1 2 in d 2 the method applies to nearest and non nearest neighbor asymmetric exclusion processes

numerical study of a non equilibrium interface model we have carried out extensive computer simulations of one dimensional models related to the low noise solid on solid non equilibrium interface of a two dimensional anchored toom model with unbiased and biased noise for the unbiased case the computed fluctuations of the interface in this limit provide new numerical evidence for the logarithmic correction to the subnormal l frac 1 2 variance which was predicted by the dynamic renormalization group calculations on the modified edwards wilkinson equation in the biased case the simulations are in close quantitative agreement with the predictions of the collective variable approximation cva which gives the same l frac 2 3 behavior of the variance as the kpz equation

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

dynamics of an anchored toom interface at a sufficiently low noise level the two dimensional toom model north east center majority vote with small errors has two stationary states the authors study the statistical properties of interfaces between these phases with particular attention to a stationary interface maintained in the third quadrant by mixed boundary conditions the fluctuations in this interface are found numerically to be much smaller than in equilibrium interfaces they have exponents 1 4 or 1 3 depending on the symmetry rather than 1 2 the correlations exhibit long range behaviour reminiscent of self organized criticality they construct several approximate theories of the interface which reproduce this behaviour at least qualitatively the most accurate of these leads to a novel nonlinear partial differential equation for the asymptotic probability distribution of the fluctuations along the interface

long time behavior for a nonlinear fourth order parabolic equation we study the asymptotic behavior of solutions of the initial boundary value problem with periodic boundary conditions for a fourth order nonlinear degenerate diffusion equation with a logarithmic nonlinearity for strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity these results are derived by analyzing the equation verified by the logarithm of the solution

on the interface between two growing eden clusters the authors have measured the roughening exponent chi at the interface between a pair of growing eden clusters their numerical results suggest that chi differs from the corresponding exponent for a freely growing interface and that it depends on the initial geometry of the clusters which grow into a vacant sector with an angle theta for theta 180

time decay and spectral kernel asymptotics for quantum systems in r3 defined by a hamiltonian h given as the sum of the negative laplacian perturbed by a real valued potential v x the large time behavior of the fundamental solution of the time dependent schrodinger equation is investigated for a suitably restricted class of potentials that have algebraic decay as x the continuous spectrum portion of the fundamental solution is characterized by an asymptotic expansion as t which is uniform in compact subsets of r3 r3 these results are then applied to derive the large energy asymptotic expansions of the spectral kernel associated with h

directed polymers and the quantum toda lattice we characterize the law of the partition function of a brownian directed polymer model in terms of a diffusion process associated with the quantum toda lattice the proof is via a multidimensional generalization of a theorem of matsumoto and yor concerning exponential functionals of brownian motion it is based on a mapping which can be regarded as a geometric variant of the rsk correspondence

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

the wasserstein gradient flow of the fisher information and the quantum drift diffusion equation we prove the global existence of non negative variational solutions to the drift diffusion evolution equation partial t u div left u mathrm d left 2 frac delta sqrt u sqrt u f right right 0 under variational boundary condition despite the lack of a maximum principle for fourth order equations non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation which is the gradient flow of the perturbed fisher information functional fancyscript f f u frac 12 int left mathrm d log u right 2 u dx int fu dx with respect to the kantorovich rubinstein wasserstein distance between probability measures we also study long time behavior of the solutions proving their exponential decay to the equilibrium state g e v characterized by delta v frac12 left mathrm d v right 2 f quad int rm e v dx int u 0 dx when the potential v is uniformly convex in this case the functional fancyscript f f coincides with the relative fisher information fancyscript f f u frac12 fancyscript i u g int left mathrm d log u g right 2u dx

order of current variance and diffusivity in the asymmetric simple exclusion process we prove that the variance of the current across a characteristic is of order t 2 3 in a stationary asymmetric simple exclusion process and that the diffusivity has order t 1 3 the proof proceeds via couplings to show the corresponding moment bounds for a second class particle

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

from equilibrium spin models to probabilistic cellular automata the general equivalence betweend dimensional probabilistic cellular automata pca and d 1 dimensional equilibrium spin models satisfying a disorder condition is first described in a pedagogical way and then used to analyze the phase diagrams the critical behavior and the universality classes of some automata diagrammatic representations of time dependent correlation functions of pca are introduced two important classes of pca are singled out for which these correlation functions simplify 1 quasi hamiltonian automata which have a current carrying steady state and for which some correlation functions are those of ad dimensional static model pca satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes 2 linear and more generally affine pca for which the diagrammatics reduces to a random walk problem closely related to d 1 dimensional directed saws both problems display a critical behavior with mean field exponents in any dimension the correlation length and effective velocity of propagation of excitations can be calculated for affine pca as is shown on an explicitd 1 example we conclude with some remarks on nonlinear pca for which the diagrammatics is related to reaction diffusion processes and which belong in some cases to the universality class of reggeon field theory

nearest neighbor markov dynamics on macdonald processes abstract macdonald processes are certain probability measures on two dimensional arrays of interlacing particles introduced by borodin and corwin in 7 they are defined in terms of nonnegative specializations of the macdonald symmetric functions and depend on two parameters q t 0 1 our main result is a classification of continuous time nearest neighbor markov dynamics on the space of interlacing arrays that act nicely on macdonald processes the classification unites known examples of such dynamics and also yields many new ones when t 0 one dynamics leads to a new integrable interacting particle system on the one dimensional lattice which is a q deformation of the pushtasep long range tasep when q t the macdonald processes become the schur processes of okounkov and reshetikhin 41 in this degeneration we discover new robinson schensted type correspondences between words and pairs of young tableaux that govern some of our dynamics

schrodinger semigroups for vector fields suppose h is the hamiltonian that generates time evolution in an n body spin dependent nonrelativistic quantum system if r is the total number of independent spin components and the particles move in three dimensions then the hamiltonian h is an r r matrix operator given by the sum of the negative laplacian x on the d 3n dimensional euclidean space rd plus a hermitian local matrix potential w x uniform higher order asymptotic expansions are derived for the time evolution kernel the heat kernel and the resolvent kernel these expansions are respectively for short times high temperatures and high energies explicit formulas for the matrix valued coefficient functions entering the asymptotic expansions are determined all the asymptotic expansions are accompanied by bounds for their respective error terms these results are obtained for the class of potentials defined as the fourier image of bounded complex valued matrix measures this class is suitable for the n body problem since interactio

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

initial time layer problem for quantum drift diffusion model we study the initial time layer problem to the quantum drift diffusion model the limit of vanishing debye length is established by entropy method and compactness argument as a byproduct the global weak solution is also obtained for this model

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

schrodinger spectral kernels high order asymptotic expansions the large energy behavior of the spectral kernel for the n body schrodinger hamiltonian is obtained in a setting of a d dimensional euclidean space without boundaries the schrodinger hamiltonian h is the sum of the negative laplacian plus a real valued local potential v x the class of potentials studied is the family of bounded and continuous functions that are formed from the fourier transforms of complex bounded measures these potentials are suitable for the n body problem since they do not necessarily decrease as x let e x y r be the family of spectral kernels generated by h in the limit explicit higher order asymptotic expansions are obtained for e x y and its associated riesz means the asymptotic expansion is uniform in x and y and is accompanied by estimates of the error term

a determinantal formula for the goe tracy widom distribution investigating the long time asymptotics of the totally asymmetric simple exclusion process sasamoto obtains rather indirectly a formula for the tracy widom distribution for the gaussian orthogonal ensemble we establish that his novel formula indeed agrees with more standard expressions

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

large time asymptotics of solutions to the generalized korteweg de vries equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized korteweg de vries gkdv equationut u 1 u x 13uxxx 0 wherex t rwhen the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time more precisely we show that the solutionu t satisfies the decay estimate u t l c 1 t 1 3 1 1 for 4 uux t l ct 2 3 1 t 1 3and using these estimates we prove the existence of the scattering stateu l2such that u t u t u l2 ct 3 3for any small initial data belonging to the weighted sobolev spaceh1 1 f l2 1 x 2 1 2 1 2x 1 2 f l2 whereu t is the airy free evolution group

a case study of an exactly solvable heat kernel the heat kernel k s tr e sh lambda e sh 0 is determined exactly for the operator h lambda delta x2 lambda lambda 1 sech2 x the behaviour of k s is extracted for both large and small s it is shown that for all s infinity k s is a continuous function of lambda although new bound states are formed when lambda passes through the positive integers this implies that the scattering states provide a discontinuous contribution to the heat kernel such that the sum of bound and scattering contributions is continuous for small s a general expression is derived for the coefficients km in the small s expansion of the heat kernel k s sigma kmsm 12 with en lambda denoting the eigenvalues of h lambda the spectral function n e sigma theta e en lambda theta e en 0 is found it is proved for general potentials that the coefficients of the large e expansion of n e nm given by n e sigma nme m 12 are related to those of the small s expansion of k s by nm km gamma 3 2 m and this is demonstrated explicitly for h lambda given above a discussion is given on the use of the small s expansion to reproduce k s

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

an extension of the derrida lebowitz speer spohn equation we show how the derivation of the derrida lebowitz speer spohn equation can be prolonged to obtain a new equation generalizing the models obtained in the paper by these authors we then investigate its properties from both an analytical and numerical perspective specifically a numerical method is presented to approximate solutions of the prolonged equation using this method we investigate the relationship between the solutions of the prolonged equation and the tracy widom goe distribution

time evolution kernels uniform asymptotic expansions for a wide class of self adjoint schrodinger hamiltonians a detailed description of the time evolution kernel is obtained in a setting of a d dimensional euclidean space without boundaries the schrodinger hamiltonian h is the sum of the negative laplacian plus a real valued local potential v x the class of potentials studied is the family of bounded and continuous functions that are formed from the fourier transforms of complex bounded measures these potentials are suitable for the n body problem since they do not necessarily decrease as x an asymptotic expansion in the complex parameter z around z 0 is derived for the family of kernels uz x y corresponding to the analytic semigroup e zh re z 0 which is uniform in the coordinate variables x and y the asymptotic expansion has a simple semiclassical interpretation furthermore an explicit bound for the remainder term in the asymptotic expansion is found the expansion and the remainder term bound continue to the time axis boundary z it