a sampling theorem on homogeneous manifolds we consider a generalization of entire functions of spherical exponential type and lagrangian splines on manifolds an analog of the paleywiener theorem is given we also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points the main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using lagrangian splines

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

tropical combinatorics and whittaker functions we establish a fundamental connection between the geometric robinson schensted knuth rsk correspondence and gl n r whittaker functions analogous to the well known relationship between the rsk correspondence and schur functions this gives rise to a natural family of measures associated with gl n r whittaker functions which are the analogues in this setting of the schur measures on integer partitions the corresponding analogue of the cauchy littlewood identity can be seen as a generalization of an integral identity for gl n r whittaker functions due to bump and stade as an application we obtain an explicit integral formula for the laplace transform of the law of the partition function associated with a 1 dimensional directed polymer model with log gamma weights recently introduced by one of the authors

dyson s nonintersecting brownian motions with a few outliers consider n nonintersecting brownian particles on r dyson brownian motions all starting from the origin at time t 0 and forced to return to x 0 at time t 1 for large n the average mean density of particles has its support for each 0 0 does it affect the brownian fluctuations along the curve c for large n in this paper we show that no new process appears as long as one considers points y t is an element of c such that 0 infinity this kernel is closely related to one found by baik ben arous and peche in the context of multivariate statistics c 2008 wiley periodicals inc

arctic boundaries of the ice model on three bundle domains in this paper we consider the six vertex model at ice point on an arbitrary three bundle domain which is a generalization of the domain wall ice model on the square or equivalently of a uniformly random alternating sign matrix we show that this model exhibits the arctic boundary phenomenon whose boundary is given by a union of explicit algebraic curves this was originally predicted by colomo and sportiello j stat phys 164 1488 1523 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries called the geometric tangent method our proof uses a probabilistic analysis of non crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case which might be of independent interest

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

spectral approximation of quadratic operator polynomials arising in photonic band structure calculations galerkin spectral approximation theory for non self adjoint quadratic operator polynomials with periodic coefficients is considered the main applications are complex band structure calculations in metallic photonic crystals periodic waveguides and metamaterials we show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part the spectral problem is equivalent to a non compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator convergence rates of finite element discretizations are considered and numerical experiments with the p p version and the h h version of the finite element method confirm the theoretical convergence rates

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

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

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

a path transformation for random walks and the robinson schensted correspondence the author and marc yor recently introduced a path transformation g k with the property that for x belonging to a certain class of random walks on z k the transformed walk g k x has the same law as the original walk conditioned never to exit the weyl chamber x x 1 less than or equal to less than or equal to x k in this paper we show that g k is closely related to the robinson schensted algorithm and use this connection to give a new proof of the above representation theorem the new proof is valid for a larger class of random walks and yields additional information about the joint law of x and g k x the corresponding results for the brownian model are recovered by donsker s theorem these are connected with hermitian brownian motion and the gaussian unitary ensemble of random matrix theory the connection we make between the path transformation g k and the robinson schensted algorithm also provides a new formula and interpretation for the latter this can be used to study properties of the robinson schensted algorithm and moreover extends easily to a continuous setting

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

fields over unsharp coordinates it has been shown that space time coordinates can exhibit only very few types of short distance structures if described by linear operators they can be continuous discrete or unsharp in one of two ways in the literature various quantum gravity models of space time at short distances point towards one of these two types of unsharpness here we investigate the properties of fields over such unsharp coordinates we find that these fields are continuous but possess only a finite density of degrees of freedom similar to fields on lattices as a special case we recover the shannon sampling theorem of information theory

conditioned random walks and the rsk correspondence we consider the stochastic evolution of three variants of the rsk algorithm giving both analytic descriptions and probabilistic interpretations symmetric functions play a key role and the probabilistic interpretations are obtained by elementary doob hunt theory in each case the evolution of the shape of the tableau obtained via the rsk algorithm can be interpreted as a conditioned random walk this is intuitively appealing and can be used for example to obtain certain relationships between orthogonal polynomial ensembles in a certain scaling limit there is a continous version of the rsk algorithm which inherits much of the structure exhibited in the discrete settings intertwining relationships between conditioned and unconditioned random walks are also given in the continuous limit these are related to the harish chandra itzyksen zuber integral

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

o connell s process as a vicious brownian motion vicious brownian motion is a diffusion scaling limit of fisher s vicious walk model which is a system of brownian particles in one dimension such that if two motions meet they kill each other we consider the vicious brownian motions conditioned never to collide with each other and call it noncolliding brownian motion this conditional diffusion process is equivalent to the eigenvalue process of the hermitian matrix valued brownian motion studied by dyson j math phys 3 1191 1962 recently o connell ann probab to be published introduced a generalization of the noncolliding brownian motion by using the eigenfunctions the whittaker functions of the quantum toda lattice in order to analyze a directed polymer model in 1 1 dimensions we consider a system of one dimensional brownian motions with a long ranged killing term as a generalization of the vicious brownian motion and construct the o connell process as a conditional process of the killing brownian motions to survive forever

uncertainty relation in quantum mechanics with quantum group symmetry the commutation relations uncertainty relations and spectra of position and momentum operators were studied within the framework of quantum group symmetric heisenberg algebras and their bargmann fock representations as an effect of the underlying noncommutative geometry a length and a momentum scale appear leading to the existence of nonzero minimal uncertainties in the positions and momenta the usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

an alternative approach to the teaching of rank basis and dimension abstract it is the author s contention that the time now spent in a linear algebra course proving results about linear independence spanning and the like is better spent on deepening the understanding of the basic concepts of maps such as the properties of being 1 1 onto invertible further the time now spent on discussing sequences of vectors is better spent discussing linear maps whose domain is some coordinate space the key idea is to make the rank of a linear map the basic concept

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

diffusions on symmetric spaces of type a iii and random matrix theories for rectangular matrices isotropic diffusion processes on cosets and and their zero curvature limit are studied from a unified viewpoint as indicated in our previous works the projection of the fokker planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the laplace beltrami operator by taking the zero curvature limit an integral which is a natural extension of the itzykson zuber integral to the rectangular matrices is explicitly evaluated the probability density function obtained from the diffusion on is studied in detail which can be applied to the quantum transport problem the explicit expressions for the probability density function in the metallic and insulating regimes are obtained for the metallic regime the integral representation for the hypergeometric function is used and the results are exact furthermore by using the orthogonal polynomial method n point correlation functions are obtained exactly for arbitrary n

crossover distributions at the edge of the rarefaction fan we consider the weakly asymmetric limit of simple exclusion process with drift to the left starting from step bernoulli initial data with so that macroscopically one has a rarefaction fan we study the fluctuations of the process observed along slopes in the fan which are given by the hopf cole solution of the kardar parisi zhang kpz equation with appropriate initial data for slopes strictly inside the fan the initial data is a dirac delta function and the one point distribution functions have been computed in comm pure appl math 64 2011 466 537 and nuclear phys b 834 2010 523 542 at the edge of the rarefaction fan the initial data is one sided brownian we obtain a new family of crossover distributions giving the exact one point distributions of this process which converge as t to those of the airy a2 bm process as an application we prove moment and large deviation estimates for the equilibrium hopf cole solution of kpz these bounds rely on the apparently new observation that the fkg inequality holds for the stochastic heat equation finally via a feynman kac path integral the kpz equation also governs the free energy of the continuum directed polymer and thus our formula may also be interpreted in those terms

transversal fluctuations of the asep stochastic six vertex model and hall littlewood gibbsian line ensembles we consider the asep and the stochastic six vertex model started with step initial data after a long time t it is known that the one point height function fluctuations for these systems are of order t1 3 we prove the kpz prediction of t2 3 scaling in space namely we prove tightness and brownian absolute continuity of all subsequential limits as t goes to infinity of the height function with spatial coordinate scaled by t2 3 and fluctuations scaled by t1 3 the starting point for proving these results is a connection discovered recently by borodin bufetov wheeler between the stochastic six vertex height function and the hall littlewood process a certain measure on plane partitions interpreting this process as a line ensemble with a gibbsian resampling invariance we show that the one point tightness of the top curve can be propagated to the tightness of the entire curve

paths in weyl chambers and random matrices baryshnikov 4 and gravner tracy widom 15 have shown that the largest eigenvalue of a random matrix of the g u e of order d has the same distribution as

a multi layer extension of the stochastic heat equation motivated by recent developments on solvable directed polymer models we define a multi layer extension of the stochastic heat equation involving non intersecting brownian motions by developing a connection with darboux transformations and the two dimensional toda equations we conjecture a markovian evolution in time for this multi layer process as a first step in this direction we establish an analogue of the karlin mcgregor formula for the stochastic heat equation and use it to prove a special case of this conjecture

transition between airy1 and airy2 processes and tasep fluctuations we consider the totally asymmetric simple exclusion process a model in the kpz universality class we focus on the fluctuations of particle positions starting with certain deterministic initial conditions for large time t one has regions with constant and linearly decreasing density the fluctuations on these two regions are given by the airy1 and airy2 processes whose one point distributions are the goe and gue tracy widom distributions of random matrix theory in this paper we analyze the transition region between these two regimes and obtain the transition process its one point distribution is a new interpolation between goe and gue edge distributions 2007 wiley periodicals inc

quantum sl_2 infinite curvature and pitman s 2m x theorem the classical theorem by pitman states that a brownian motion minus twice its running infimum enjoys the markov property on the one hand biane understood that pitman s theorem is intimately related to the representation theory of the quantum group mathcal u q left mathfrak sl 2 right in the so called crystal regime q rightarrow 0 on the other hand bougerol and jeulin showed the appearance of exactly the same pitman transform in the infinite curvature limit r rightarrow infty of a brownian motion on the hyperbolic space mathbb h 3 sl 2 mathbb c su 2 this paper aims at understanding this phenomenon by giving a unifying point of view in order to do so we exhibit a presentation mathcal u q hbar left mathfrak sl 2 right of the jimbo drinfeld quantum group which isolates the role of curvature r and that of the planck constant hbar the simple relationship between parameters is q e r the semi classical limits hbar rightarrow 0 are the poisson lie groups dual to sl 2 mathbb c with varying curvatures r in mathbb r we also construct classical and quantum random walks drawing a full picture which includes biane s quantum walks and the construction of bougerol jeulin taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory hbar 0 and to the bougerol jeulin construction in the classical world hbar 0 all these results are neatly in accordance with the philosophy of kirillov s orbit method

generalization of the bauer fike theorem the bauer fike theorem for diagonalizable matrices is generalized to cases where i non diagonalizable matrices or ii only part of the spectrum is considered

renormalization fixed point of the kpz universality class the one dimensional kardar parisi zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents these exponents lead to a natural renormalization rescaling on the space of such evolving interfaces we introduce and describe the renormalization fixed point of the kardar parisi zhang universality class in terms of a random nonlinear semigroup with stationary independent increments and via a variational formula furthermore we compute a plausible formula the exact transition probabilities using replica bethe ansatz the semigroup is constructed from the airy sheet a four parameter space time field which is the airy 2 process in each of its two spatial coordinates minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential at present the results we provide do not have mathematically rigorous proofs and they should at most be considered proposals

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

weak type 1 1 estimates for heat kernel maximal functions on lie groups for a lie group g with left invariant haar measure and associated lebesgue spaces lp g we consider the heat kernels pt t o arising from a right invariant laplacian a on g that is u t pt f solves the heat equation a ot a u 0 with initial condition u o f we establish weak type 1 1 estimates for the maximal operator xf ji f supt 0 pt f 1 and for related hardy littlewood maximal operators in a variety of contexts namely for groups of polynomial growth and for a number of classes of iwasawa an groups we also study the local maximal operator x0 of sup0 o t ipt f i and related hardy littlewood operators for all lie groups

the class of n entire operators we introduce a classification of simple regular closed symmetric operators with deficiency indices 1 1 according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to m g krein we show that these classes of operators have several distinctive properties some of them related to the spectra of their canonical self adjoint extensions in particular we provide necessary and sufficient conditions on the spectra of two canonical self adjoint extensions of an operator for it to belong to one of our classes our discussion is based on some recent results in the theory of de branges spaces

kpz line ensemble for each t ge 1 we construct an mathbb n indexed ensemble of random continuous curves with three properties 1 the lowest indexed curve is distributed as the time t hopf cole solution to the kardar parisi zhang kpz stochastic partial differential equation with narrow wedge initial data 2 the entire ensemble satisfies a resampling invariance which we call the mathbf h brownian gibbs property with mathbf h x e x 3 increments of the lowest indexed curve when centered by t 24 and scaled down vertically by t 1 3 and horizontally by t 2 3 remain uniformly absolutely continuous i e have tight radon nikodym derivatives with respect to brownian bridges as time t goes to infinity this construction uses as inputs the diffusion that o connell discovered ann probab 40 437 458 2012 in relation to the o connell yor semi discrete brownian polymer the convergence result of moreno flores et al in preparation of the lowest indexed curve of that diffusion to the solution of the kpz equation with narrow wedge initial data and the one point distribution formula proved by amir et al commun pure appl math 64 466 537 2011 for the solution of the kpz equation with narrow wedge initial data we provide four main applications of this construction 1 uniform as t goes to infinity brownian absolute continuity of the time t solution to the kpz equation with narrow wedge initial data even when scaled vertically by t 1 3 and horizontally by t 2 3 2 universality of the t 1 3 one point vertical fluctuation scale for the solution of the kpz equation with general initial data 3 concentration in the t 2 3 scale for the endpoint of the continuum directed random polymer 4 exponential upper and lower tail bounds for the solution at fixed time of the kpz equation with general initial data

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

applications of krein s theory of regular symmetric operators to sampling theory the classical kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae one major task in sampling theory is to find concrete non trivial realizations of this theorem in this paper we provide a new approach to this subject on the basis of krein s theory of representation of simple regular symmetric operators having deficiency indices 1 1 we show that the resulting sampling formulae have the form of lagrange interpolation series we also characterize the space of function reconstructible by our sampling formulae our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity

covariant information density cutoff in curved space time in information theory the link between continuous information and discrete information is established through well known sampling theorems sampling theory explains for example how frequency filtered music signals are reconstructible perfectly from discrete samples in this letter sampling theory is generalized to pseudo riemannian manifolds this provides a new set of mathematical tools for the study of space time at the planck scale theories formulated on a differentiable space time manifold can be equivalent to lattice theories there is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity

discrete perturbation estimates for eigenpairs of fredholm operator valued functions we present perturbation estimates for eigenvalue and eigenvector approximations for a class of fredholm operator valued functions our approach is based on perturbation estimates for the generalized resolvents and the exponential convergence of the contour integration by the trapezoidal rule we use discrete residual functions to estimate the resolvents a posteriori numerical experiments are also presented

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

analysis of superoscillatory wave functions surprisingly differentiable functions are able to oscillate arbitrarily faster than their highest fourier component would suggest the phenomenon is called superoscillation recently a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter intuitive physical effects following up on this work we here present more general methods which allow the calculation of superoscillatory wave functions with custom designed physical properties we give concrete examples and we prove results about the limits to superoscillatory behavior we also give a simple and intuitive new explanation for the exponential computational cost of superoscillations

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

laplace s method for iterated complex brownian integrals a kind of laplace s method is developped for iterated stochastic integrals where integrators are complex standard brownian motions then it is used to extend properties of bougerol and jeulin s path transform in the random case when simple representations of complex semisimple lie algebras are not supposed to be minuscule

anomalous shock fluctuations in tasep and last passage percolation models we consider the totally asymmetric simple exclusion process with initial conditions and or jump rates such that shocks are generated if the initial condition is deterministic then the shock at time t will have a width of order t 1 3 we determine the law of particle positions in the large time limit around the shock in a few models in particular we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the airy 1 process the limiting distribution is a product of two distribution functions which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics we show that the result generalizes to generic last passage percolation models

infinite dimensional stochastic differential equations for dyson s model in this paper we show the strong existence and the pathwise uniqueness of an infinite dimensional stochastic differential equation sde corresponding to the bulk limit of dyson s brownian motion for all beta ge 1 our construction applies to an explicit and general class of initial conditions including the lattice configuration x i mathbb z and the sine process we further show the convergence of the finite to infinite dimensional sde this convergence concludes the determinantal formula of katori and tanemura commun math phys 293 2 469 497 2010 for the solution of this sde at beta 2

on the chaotic character of the stochastic heat equation before the onset of intermitttency we consider a nonlinear stochastic heat equation tu 1 xxu u xtw where xtw denotes space time white noise and r r is lipschitz continuous we establish that at every xed time t 0 the global behavior of the solution depends in a critical manner on the structure of the initial function u0 under suitable technical conditions on u0 and supx2rut x is a s nite when u0 has compact support whereas with probability one lim supjxj 1ut x logjxj 1 6 0 when u0 is bounded uniformly away from zero the mentioned sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at xed times well before the onset of intermittency

on the chaotic character of the stochastic heat equation ii consider the stochastic heat equation partial t u varkappa 2 delta u sigma u dot f where the solution u u t x is indexed by t x in 0 infty times bf r d and dot f is a centered gaussian noise that is white in time and has spatially correlated coordinates we analyze the large x fixed t behavior of the solution u in different regimes thereby study the effect of noise on the solution in various cases among other things we show that if the spatial correlation function f of the noise is of riesz type that is f x propto x alpha then the fluctuation exponents of the solution are psi for the spatial variable and 2 psi 1 for the time variable where psi 2 4 alpha moreover these exponent relations hold as long as alpha in 0 d wedge 2 that is precisely when dalang s theory dalang electron j probab 4 paper no 6 29 1999 implies the existence of a solution to our stochastic pde these findings bolster earlier physical predictions kardar et al phys rev lett 58 20 889 892 1985 kardar and zhang phys rev lett 58 20 2087 2090 1987

transversal fluctuations for increasing subsequences on the plane consider a realization of a poisson process in 2 with intensity 1 and take a maximal up right path from the origin to n n consisting of line segments between the points where maximal means that it contains as many points as possible the number of points in such a path has fluctuations of order n where 1 3 bdj here we show that typical deviations of a maximal path from the diagonal x y is of order n with 2 3 this is consistent with the scaling identity 2 1 which is believed to hold in many random growth models