the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

asymptotic behavior of stochastic fitzhugh nagumo systems on unbounded thin domains abstract this article is concerned with the asymptotic behavior of solutions of a class of non autonomous stochastic fitzhugh nagumo systems driven by multiplicative white noise on unbounded thin domains the existence and uniqueness of tempered pullback random attractors are proved for the systems defined on n 1 dimensional unbounded narrow domains we also establish the upper semicontinuity of these attractors when a family of n 1 dimensional unbounded thin domains collapses onto an n dimensional unbounded domain a cut off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non compactness of sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

universal behavior for averages of characteristic polynomials at the origin of the spectrum it has been shown by strahov and fyodorov that averages of products and ratios of characteristic polynomials corresponding to hermitian matrices of a unitary ensemble involve kernels related to orthogonal polynomials and their cauchy transforms we will show that for the unitary ensemble open image in new window hermitian matrices these kernels have universal behavior at the origin of the spectrum as n in terms of bessel functions our approach is based on the characterization of orthogonal polynomials together with their cauchy transforms via a matrix riemann hilbert problem due to fokas its and kitaev and on an application of the deift zhou steepest descent method for matrix riemann hilbert problems to obtain the asymptotic behavior of the riemann hilbert problem

green currents for holomorphic automorphisms of compact kahler manifolds and some additional regularity properties very likely the currents t will describe the distribution of invariant manifolds of codimension s corresponding to the small est lyapounov exponents let dp denote the dynamical degree of order p of it describes the growth under iteration of the volume of p dimensional manifolds when d 1 it is natural to introduce first a positive closed 1 l current as ifnyoj 7 lim

one dimensional random attractor and rotation number of the stochastic damped sine gordon equation abstract this paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine gordon equation with homogeneous neumann boundary condition it is shown that for any positive damping and diffusion coefficients the equation possesses a random attractor and when the damping and diffusion coefficients are sufficiently large the random attractor is a one dimensional random horizontal curve regardless of the strength of noise hence its dynamics is not chaotic it is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

on random matrices from the compact classical groups if m is a matrix taken randomly with respect to normalized haar measure on u n o n or sp n then the real and imaginary parts of the random variables tr mk k 1 converge to independent normal random variables with mean zero and variance k 2 as the size n of the matrix goes to infinity for the unitary group this is a direct consequence of the strong szeg6 theorem for toeplitz determinants we will prove a conjecture of diaconis

random dynamics of fractional stochastic reaction diffusion equations on rn without uniqueness this paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction diffusion equations on rn with continuous but not necessarily differentiable nonlinear drift terms which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation we first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero the measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions this paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction diffusion equations on rn with continuous but not necessarily differentiable nonlinear drift terms which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation we first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero the measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

asymptotic behavior of non autonomous fractional stochastic reaction diffusion equations abstract we first apply the galerkin method to prove the existence and uniqueness of solutions for a class of non autonomous fractional reaction diffusion equations driven by multiplicative noise we then establish the existence and uniqueness of tempered pullback random attractors for the equations in an appropriate hilbert space the upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero

integration by parts formula for regional fractional laplacian we obtain the integration by parts formula for the regional fractional laplacian which are generators of symmetric stable processes on a subset of mathbb r n 0 2 in this formula a local operator appears on the boundary connected with the regional fractional laplacian on domain hence this formula can be understood as the green formula for the regional fractional laplacian the similar integration by parts formula is also given for more general nonlocal operators the reflected stable like processes are studied

henon like mappings in c we study the dynamics of a class of nonalgebraic holomorphic diffeomorphisms topo logical analogues in the unit bidisk of complex h6non mappings in c2 in particular a dynamical degree is defined which is related to topological entropy and we construct stable unstable invariant currents and prove there is a unique mixing measure of maximal entropy with product structure 1 introduction one of the most famous 2 dimensional dynamical systems is smale s horseshoe it is a basic example of a simple map with complicated dynamics and has served as a guideline for the whole modern development of discrete time dynamical systems a very natural problem in this context is to understand the dynamical phenomena occuring when uniform hyperbolicity of the horseshoe degenerates this was a motivation for the study of the so called henon map

orthogonal polynomials asymptotics and heun equations the painleve equations arise from the study of hankel determinants generated by moment matrices whose weights are expressed as the product of classical weights multiplied by suitable deformation factors usually dependent on a time variable t from ladder operators see a magnus j comput appl math 57 1 2 215 237 1995 one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions the painleve and related functions appear as the residues of these rational functions we will be interested in the situation when n the order of the hankel matrix and also the degree of the polynomials pn x orthogonal with respect to the deformed weights gets large we show that the second order linear differential equations satisfied by pn x are particular cases of heun equations when n is large in some sense monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of heun equations heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations in this paper we look at three types of weights the jacobi type the laguerre type and the weights deformed by the indicator function of a b x and the step function x in particular we consider the following jacobi type weights 1 1 x 1 x e tx x 0 1 t 0 1 2 x 1 x e t x x 0 1 t 0 1 3 1 x2 1 k2x2 x 1 1 0 k2 0 1 the laguerre type weights 2 1 x x t e x x 0 t 0 2 2 x e x t x x 0 t 0 and another type of deformation when the classical weights are multiplied by a b x or x 3 1 e x2 1 a a x x r a 0 3 2 1 x2 1 a a x x 1 1 a 0 1 0 3 3 x e x a b x t x 0 t 0 a 0 a b 0 the weights mentioned above were studied in a series of papers related to the deformation of classical weights

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

anomalous diffusion in view of einstein s 1905 theory of brownian motion abstract einstein s theory of brownian motion is revisited in order to formulate a generalized kinetic theory of anomalous diffusion it is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed the fractional derivative naturally appears in the diffusion equation this is the first demonstration of the physical origin of the fractional derivative in marked contrast to the usual phenomenological introduction of it it is also examined if einstein s approach can be generalized to nonlinear kinetic theory to derive a generalization of the porous medium type equation by the appropriate use of the escort distribution

limiting dynamics for stochastic wave equations abstract in this paper relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered by introducing almost surely d contracting property for random dynamical systems we obtain a global random attractor of the stochastic wave equation u t t u t u f u w endowed with dirichlet boundary condition for any 0 1 the upper semicontinuity of this global random attractor and the global attractor of the heat equation z t z f z 0 with dirichlet boundary condition as goes to zero is investigated furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

laguerre ensembles of random matrices nonuniversal correlation functions the eigenvalue correlation functions of the laguerre ensembles exponential ensembles of real symmetric and self dual quaternion random matrices are calculated it is shown that in certain intervals of the spectrum the renormalized correlations are not universal a physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given

a riemann hilbert approach to asymptotic questions for orthogonal polynomials abstract a few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials in this paper we give an overview of our method and review the results which have been obtained in deift et al internat math res notices 1997 759 comm pure appl math 52 1999 1491 1335 deift orthogonal polynomials and random matrices a riemann hilbert approach courant lecture notes vol 3 new york university 1999 kriecherbauer and mclaughlin internat math res notices 1999 299 and baik et al j amer math soc 12 1999 1119 we mainly consider orthogonal polynomials with respect to weights on the real line which are either 1 freud type weights d x e q x d x q polynomial or q x x 0 or 2 varying weights d n x e nv x d x v analytic lim x v x log x we obtain plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae with error estimates for the leading coefficients for the recurrence coefficients and for the zeros of the orthogonal polynomials our proof starts from an observation of fokas et al comm math phys 142 1991 313 that the orthogonal polynomials can be determined as solutions of certain matrix valued riemann hilbert problems we analyze the riemann hilbert problems by a steepest descent type method introduced by deift and zhou ann math 137 1993 295 and further developed in deift and zhou comm pure appl math 48 1995 277 and deift et al proc nat acad sci usa 95 1998 450 a crucial step in our analysis is the use of the well known equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

hermite pade approximants for a pair of cauchy transforms with overlapping symmetric supports hermite pade approximants of type ii are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order we are interested in the situation when the approximated vector is given by a pair of cauchy transforms of smooth complex measures supported on the real line the convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals angelesco systems or two intervals that coincide under the condition that the ratio of the measures is a restriction of the cauchy transform of a third measure nikishin systems in this work we consider the case where the supports form two overlapping intervals in a symmetric way and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap we derive szeg type formulae for the asymptotics of the approximants identify the convergence and divergence domains the divergence domains appear for angelesco systems but are not present for nikishin systems and show the presence of overinterpolation a feature peculiar for nikishin systems but not for angelesco systems our analysis is based on a riemann hilbert problem for multiple orthogonal polynomials the common denominator 2016 wiley periodicals inc

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

dynamics of automorphisms on compact kahler manifolds we study holomorphic automorphisms on compact kahler manifolds having simple actions on the hodge cohomology ring we show for such automorphisms that the main dynamical green currents admit complex laminar structures woven currents and the green measure is the unique invariant probability measure of maximal entropy

the existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion abstract in this paper we investigate the existence uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional brownian motion b q h t d x t a x t f t x t d t g t d b q h t with hurst parameter h 1 2 1 we also consider the existence of weak solutions

a variational model for dislocations in the line tension limit we study the interaction of a singularly perturbed multiwell energy with an anisotropic nonlocal regularizing term of h1 2 type and a pinning condition this functional arises in a phase field model for dislocations which was recently proposed by koslowski cuitino and ortiz but it is also of broader mathematical interest in the context of the dislocation model we identify the limit of the energy in all scaling regimes for the number n of obstacles the most interesting regime is n ln where is a nondimensional length scale related to the size of the crystal lattice in this case the limiting model is of line tension type one important feature of our model is that the set of energy wells is periodic and hence not compact thus a key ingredient in the proof is a compactness estimate up to a single translation for finite energy sequences which generalizes earlier results from alberti bouchitte and seppecher for the two well problem with a h1 2 regularization

regularity forward compactness and measurability of attractors for non autonomous stochastic lattice systems abstract this paper is concerned with the regularity forward compactness and measurability of pullback random attractors of non autonomous stochastic lattice systems with multiplicative white noise as well as random coefficients in weighted spaces the existence uniqueness and forward compactness of pullback random attractors in weighted space l 2 are proved for the systems when the random coefficient in the front of the discrete laplacian and the time dependent functions satisfy certain conditions in addition this forward compact l 2 l 2 attractor is proved to be a forward compact bi spatial l 2 l 2 l p attractor that is forward compact and measurable in l 2 l p and attracts all random subset of l 2 under the topology of l 2 l p the difficulty of establishing the measurability of the attractors in l 2 l p is overcome by proving the identity of the attractors on two different universes

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

on the spectrum of two different fractional operators in this paper we deal with two non local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type more precisely for a fixed s 0 1 we consider the integral definition of the fractional laplacian given by where c n s is a positive normalizing constant and another fractional operator obtained via a spectral definition that is where e i i are the eigenfunctions and the eigenvalues of the laplace operator in with homogeneous dirichlet boundary data while a i represents the projection of u on the direction e i the aim of this paper is to compare these two operators with particular reference to their spectrum in order to emphasize their differences

reflected symmetric α stable processes and regional fractional laplacian in this paper we investigate the reflected symmetric stable processes and their generators we show that the generators are regional fractional laplacians on the closed region in the case of 1 2 their existence requires that open image in new window on the boundary among other things we obtain the integration by parts formula of the regional fractional laplacian and the semi martingale decomposition of the reflected symmetric stable processes

asymptotic behavior of non autonomous fractional stochastic p laplacian equations abstract this article is concerned with the asymptotic behavior of solutions of a wide class of non autonomous non local fractional stochastic p laplacian equations driven by multiplicative white noise the time dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p we first employ the faedo galerkin method to prove the well posedness of the equation in an appropriate hilbert space we then establish the existence uniqueness and periodicity of tempered pullback random attractors for the equations the upper semi continuity of these attractors is also derived as the density of noise tends to zero the results of this paper are new even when the stochastic equation reduces to the deterministic one

perturbed hankel determinants in this paper we compute for largen the determinant of a class of n n hankel matrices which arise from a smooth perturbation of the jacobi weight for this purpose we employ the same idea used in previous papers where the unknown determinant dn w h is compared with the known determinant dn w here w is the jacobi weight and w h where h h x x 1 1 is strictly positive and real analytic is the smooth perturbation on the jacobi weight w x 1 x 1 x applying a previously known formula on the distribution function of linear statistics we compute the large n asymptotics of dn w h and supply a missing constant of the expansion

critical edge behavior in the modified jacobi ensemble and the painleve v transcendents we study the jacobi unitary ensemble defined by the perturbed chebyshev weight w x 1 x2 12 tn2 x2 x 1 1 tn 1 the algebraic singularity coalesces with the end point of the support of the weight function namely tn 1 the hard edge as n we use the riemann hilbert approach the method of deift and zhou to obtain the asymptotics of the polynomials orthogonal with respect to w x we show that the universality property is preserved the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel the main result is on the local behavior at the edge of the spectrum the limit kernel at the edge when tn 1 o n 2 is described by the function for a specific solution of the painleve v equation we also show that when tn varies to a fixed t 1 the painleve v kernel transits to the bessel kernel j 12 on the other hand when tn approaches 1 the painleve v kernel can be approximated by another bessel kernel j 12

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

local uniformly upper semi continuity of random attractor for g navier stokes equation this paper is concerned with the locally uniform convergence from a family of pullback random attractors to a deterministic attractor we establish criteria by using joint pathwise convergence of the cocycles collective locally uniform compactness and eventually deterministic of the random attractors as an application of the abstract result it is shown that the family of random attractors for the stochastic g navier stokes equation is local uniformly upper semi continuous as the density of random noise tends to zero

asymptotic expansions for second order linear difference equations with a turning point a turning point theory is developed for the second order difference equation where the coefficients a n and b n have asymptotic expansions of the form 0 being a real number in particular it is shown how the airy functions arise in the uniform asymptotic expansions of the solutions to this three term recurrence relation as an illustration of the main result a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the freud weight exp x 4 x

regularity of random attractors for fractional stochastic reaction diffusion equations on rn abstract we investigate the regularity of random attractors for the non autonomous non local fractional stochastic reaction diffusion equations in h s r n with s 0 1 we prove the existence and uniqueness of the tempered random attractor that is compact in h s r n and attracts all tempered random subsets of l 2 r n with respect to the norm of h s r n the main difficulty is to show the pullback asymptotic compactness of solutions in h s r n due to the noncompactness of sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the wiener process we establish such compactness by the ideas of uniform tail estimates and the spectral decomposition of solutions in bounded domains

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

a riemann hilbert problem for skew orthogonal polynomials we find a local d 1 x d 1 riemann hilbert problem characterizing the skew orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d our riemann hilbert problem is similar to a local dxd riemann hilbert problem found by kuijlaars and mclaughlin characterizing the bi orthogonal polynomials this gives more motivation for finding methods to compute asymptotics of high order riemann hilbert problems and brings us closer to finding full asymptotic expansions of the skew orthogonal polynomials

asymptotics for toeplitz determinants perturbation of symbols with a gap we study the determinants of toeplitz matrices as the size of the matrices tends to infinity in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate we generalize an asymptotic expansion by widom indiana univ math j 21 277 283 1971 which was known for symbols supported on an arc we highlight applications of our results in the circular unitary ensemble and in the study of fredholm determinants associated to the sine kernel

universality for eigenvalue correlations at the origin of the spectrum we establish universality of local eigenvalue correlations in unitary random matrix ensembles frac 1 z n det m 2 alpha e n rm tr v m dm near the origin of the spectrum if v is even and if the recurrence coefficients of the orthogonal polynomials associated with x 2 e nv x have a regular limiting behavior then it is known from work of akemann et al and kanzieper and freilikher that the local eigenvalue correlations have universal behavior described in terms of bessel functions we extend this to a much wider class of confining potentials v our approach is based on the steepest descent method of deift and zhou for the asymptotic analysis of riemann hilbert problems this method was used by deift et al to establish universality in the bulk of the spectrum a main part of the present work is devoted to the analysis of a local riemann hilbert problem near the origin

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

limiting behavior of non autonomous stochastic reaction diffusion equations on unbounded thin domains this article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains we first prove the existence of tempered pullback random attractors for the equations defined on n 1 dimensional unbounded thin domains then we show the upper semicontinuity of these attractors when the n 1 dimensional unbounded thin domains collapse onto the n dimensional space rn here the tail estimates are utilized to deal with the noncompactness of sobolev embeddings on unbounded domains

dynamics of fractional stochastic reaction diffusion equations on unbounded domains driven by nonlinear noise abstract this paper is concerned with the asymptotic behavior of the solutions of the fractional reaction diffusion equations with polynomial drift terms of arbitrary order driven by locally lipschitz nonlinear diffusion terms defined on r n we first prove the well posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average we then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors we finally establish the existence of invariant measures when the diffusion terms are globally lipschitz continuous the uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non compactness of usual sobolev embeddings on unbounded domains

regularity and backward compactness of attractors for non autonomous lattice systems with random coefficients abstract we study longtime behavior for the non autonomous lattice model with multiplicative white noise and a random coefficient in the discrete laplace operator we first show existence of a bi spatial attractor when the initial space is the weighted square summation space and the terminal space is the weighted p times summation space for p 2 we then show backward compactness of the attractor in both initial and terminal spaces if the time indexed forces are backward tempered and backward null finally by proving identity of the attractors on the different universes of tempered or backward tempered sets we show measurability of the attractor in the initial space and in the terminal space respectively