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

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

colored solutions of the yang baxter equation from representations of uqgl 2 we study the hopf algebra structure and the highest weight representation of a multiparameter version of uqgl 2 the hopf algebra maps of this algebra are explicitly given we show that the multiparameter universal r matrix can be constructed directly as a quantum double intertwiner without using reshetikhin s twisting transformation we find there are two types highest weight representations for this algebra type a corresponds to the qeneric q and type b corresponds to the case that q is a root of unity when applying the representation theory to the multi parameter universal r matrix both standard and nonstandard colored solutions of the yang baxter equation are obtained

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

random matrix central limit theorems for nonintersecting random walks we consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function we prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity several limiting distributions of the walks at the mid time behave as the eigenvalues of random hermitian matrices as the dimension of the matrices grows to infinity

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

transfer matrix eigenvalues of the anisotropic multiparametric u model a multiparametric extension of the anisotropic u model is discussed which maintains integrability the r matrix solving the yang baxter equation is obtained through a twisting construction applied to the underlying u q sl 2 1 superalgebraic structure which introduces the additional free parameters that arise in the model three forms of bethe ansatz solution for the transfer matrix eigenvalues are given which we show to be equivalent

a local mesh refinement multigrid method for 3 d convection problems with strongly variable viscosity a numerical method for solving 3 d convection problems with variable viscosity in cartesian geometry is presented equations for conservation of mass momentum and energy are solved using a second order finite volume discretization in combination with a multigrid method viscosity variations of 10 orders of magnitude are considered convergence deteriorates with increasing viscosity variations but modifications of the multigrid algorithm are found to improve the robustness of the numerical method for very large viscosity contrasts an efficient and flexible local mesh refinement technique is presented which is applied to various convection problems with variable viscosity comparisons with other numerical methods reveal that accurate results are obtained even when viscosity varies strongly

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

determinantal martingales and correlations of noncolliding random walks we study the noncolliding random walk rw which is a particle system of one dimensional simple and symmetric rws starting from distinct even sites and conditioned never to collide with each other when the number of particles is finite n infty this discrete process is constructed as an h transform of absorbing rw in the n dimensional weyl chamber we consider fujita s polynomial martingales of rw with time dependent coefficients and express them by introducing a complex markov process it is a complexification of rw in which independent increments of its imaginary part are in the hyperbolic secant distribution and it gives a discrete time conformal martingale the h transform is represented by a determinant of the matrix whose entries are all polynomial martingales from this determinantal martingale representation dmr of the process we prove that the noncolliding rw is determinantal for any initial configuration with n infty and determine the correlation kernel as a function of initial configuration we show that noncolliding rws started at infinite particle configurations having equidistant spacing are well defined as determinantal processes and give dmrs for them tracing the relaxation phenomena shown by these infinite particle systems we obtain a family of equilibrium processes parameterized by particle density which are determinantal with the discrete analogues of the extended sine kernel of dyson s brownian motion model with beta 2 following donsker s invariance principle convergence of noncolliding rws to the dyson model is also discussed

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

multigrid solution of the incompressible navier stokes equations in general coordinates galerkin coarse grid approximation gca in multigrid methods is investigated for the incompressible navier stokes equations in general coordinates an efficient algorithm performing gca is presented the behavior of coarse grid matrices is studied under gca with different transfer operators for square and l shaped driven cavity problems the performance of the multigrid method using different combinations of transfer operators for the computation of coarse grid matrices and of coarse grid correction is investigated further computations are carried out in general coordinates for a channel flow problem with backward facing step in three dimensions

development of a stokes flow solver robust to large viscosity jumps using a schur complement approach with mixed precision arithmetic we develop an iterative solution technique for solving stokes flow problems with smooth and discontinuous viscosity structures using a three dimensional staggered grid finite difference discretization two preconditioned iterative methodologies are applied to the saddle point arising from the discrete stokes problem they consist of a velocity pressure coupled approach fc and a decoupled schur complement approach sc within both of these methods we utilize either the scaled bfbt or an identity matrix scaled by the local cell viscosity lv to define a preconditioner for the schur complement additionally we propose to use a mixed precision krylov kernel to improve the convergence by reducing round off error in this approach standard double precision is used during the application of the preconditioner whilst higher precision arithmetic is used to define the matrix vector product dot products and norms required by the krylov method in our krylov kernel we utilize quad precision arithmetic which is emulated via the double double precision method we consider several simplified geodynamic problems with a viscosity contrast to demonstrate the robustness and scalability of our solution methods through a careful choice of stopping conditions we are able to quantitatively compare the residuals between the sc and fc approaches we examine the trade off relationship between the number of outer iterations required for convergence and the computational cost per iteration for the each solution methods we find that it is advantageous to use the fc approach utilizing relaxed tolerances for solution of the sub problems combined with the lv preconditioner we also observed that in general the sc approach is more robust than fc and that bfbt is more robust than lv when used in our numerical experimental in addition our mixed precision method produces improved convergence rates of arnoldi type krylov subspace methods without a drastic increasing the computational time the usage of a high precision krylov kernel is found to be useful for the solver associated with the velocity sub problem

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

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

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

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

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

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation

random vicious walks and random matrices a lock step walk is a one dimensional integer lattice walk in discrete time suppose that initially there are infinitely many walkers on the nonnegative even integer sites at each moment of time every walker moves either to its left or to its right with equal probability the only constraint is that no two walkers can occupy the same site at the same time hence we describe the walk as vicious it is proved that as time tends to infinity a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the scaled largest eigenvalue of a random goe matrix goe tracy widom distribution the proof is based on the bijection between path configurations and semistandard young tableaux established recently by guttmann owczarek and viennot the distribution of semistandard young tableaux is analyzed using the hankel determinant expression for the probability obtained from the work of rains and the author the asymptotics of the hankel determinant are then obtained by applying the deift zhou steepest descent method to the riemann hilbert problem for the related orthogonal polynomials 2000 john wiley sons inc

three dimensional eulerian method for large deformation of viscoelastic fluid toward plate mantle simulation toward the unified simulation of the large deformation of a rigid viscoelastic material plate and the convection of a viscous fluid mantle an eulerian scheme with a semi lagrangian method is developed the scheme adopts the cip cslr method for advection terms of staggered grid system in three dimensions the positive transported profile of a positive quantity is assured by flux corrections in the dimensional splitting method the jaumann co rotational effect of the stress tensor is also integrated into the semi lagrangian treatment this co rotated semi lagrangian method is combined with an exponential time differencing method in the time development of the maxwell constitutive model the large time step comparable to or larger than the maxwell relaxation time is successfully realized validation tests are performed for the three dimensional rayleigh taylor instability of a viscoelastic material with jump discontinuity of the mass density and other material properties

highest weight uq sl n modules and invariant integrable n state models with periodic boundary conditions weights are computed for the bethe vectors of an rsos type model with periodic boundary conditions obeying uq sl n q exp i pi r invariance they are shown to be highest weight vectors the q dimensions of the corresponding irreducible representations are obtained

pressure based residual smoothing operators for multistage pseudocompressibility algorithms implicit residual smoothing operators for accelerating the convergence of explicit multistage artificial compressibility algorithms are developed using ideas from pressure based methods the velocity derivatives in the continuity equation and the pressure gradient terms in the momentum equations are discretized in time implicitly the discrete system of equations is linearized in time producing a block implicit operator which is approximately factorized and diagonalized via a similarity transformation the so derived diagonal operator depends only on the metrics of the geometric transformation and can thus be implemented in an efficient and straightforward manner it is combined with the standard implicit residual smoothing operator and incorporated in a four stage runge kutta algorithm also enhanced with local time stepping and multigrid acceleration linear stability analysis for the three dimensional navier stokes equations and calculations for laminar flows through curved square ducts and pipes demonstrate the damping properties and efficiency of the proposed approach particularly on large aspect ratio highly skewed meshes

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

on a quantum group invariant spin chain with non local boundary conditions we treat a one parameter family of quantum spin models which are quantum group invariant as well as periodic in a certain sense using bethe states we analyse its ground state properties and obtain the surprising result that its spin becomes non zero depending on the value of the parameter finite size corrections allow us to obtain the central charge which is also spin dependent relations to other models are mentioned

vertex centered and cell centered multigrid for interface problems abstract cell centered and vertex centered multigrid methods for solving interface problems are studied these methods differ in the location of the nodes in the grids and in the transfer operators it is shown how by means of stencil notation a compact and precise description can be given of the transfer and coarse grid operators a structured fortran description of the fundamental multigrid algorithm with only one goto statement is presented numerical results of several test problems with strong discontinuities in equation coefficients are presented storage and work requirements are discussed

random walks and random permutations a connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the gue random hermitian matrices the scaling of the distribution gives the maximum mean displacement after t time steps as 2t 1 2 with standard deviation proportional to 1 3 the exponent 1 3 is typical of a large class of two dimensional growth problems

cell centered multigrid for interface problems abstract a multigrid method is presented for cell centered discretizations of elliptic partial differential equations the method works both for smooth and strongly discontinuous coefficients even though in contrast with earlier works the prolongation and restriction operators do not depend on the equation

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

a model for the bus system in cuernavaca mexico the bus system in cuernavaca mexico and its connections to random matrix distributions have been the subject of an interesting recent study by m krbalek and p seba in 15 16 in this paper we introduce and analyse a microscopic model for the bus system we show that introducing a natural repulsion does produce random matrix distributions in natural double scaling regimes the techniques employed include non intersecting paths logarithmic potential theory determinantal point processes and asymptotic analysis of several orthogonal polynomial ensembles in addition we introduce a circular bus model and include various calculations of non crossing probabilities

q deformations of the o 3 symmetric spin 1 heisenberg chain the authors present the general expression for the spin 1 heisenberg chain invariant under the uq so 3 quantum algebra several physical and mathematical implications are discussed

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

practical use of variational principles for modeling water waves this paper describes a method for deriving approximate equations for irrotational water waves the method is based on a relaxed variational principle i e on a lagrangian involving as many variables as possible this formulation is particularly suitable for the construction of approximate water wave models since it allows more freedom while preserving the variational structure the advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters as well as arbitrary depths using subordinate constraints e g irrotationality or free surface impermeability in various combinations several model equations are derived some being well known other being new the models obtained are studied analytically and exact travelling wave solutions are constructed when possible

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

pdes for the joint distributions of the dyson airy and sine processes in a celebrated paper dyson shows that the spectrum of an n x n random hermitian matrix diffusing according to an ornstein uhlenbeck process evolves as n noncolliding brownian motions held together by a drift term the universal edge and bulk scalings for hermitian random matrices applied to the dyson process lead to the airy and sine processes in particular the airy process is a continuous stationary process describing the motion of the outermost particle of the dyson brownian motion when the number of particles gets large with space and time appropriately rescaled in this paper we answer a question posed by kurt johansson to find a pde for the joint distribution of the airy process at two different times similarly we find a pde satisfied by the joint distribution of the sine process this hinges on finding a pde for the joint distribution of the dyson process which itself is based on the joint probability of the eigenvalues for coupled gaussian hermitian matrices the pde for the dyson process is then subjected to an asymptotic analysis consistent with the edge and bulk rescalings the pdes enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t 1 and t 2 when t 2 t 1 infinity as illustrated in this paper for the airy process this paper also contains a rigorous proof that the extended hermite kernel governing the joint probabilities for the dyson process converges to the extended airy and sine kernels after the appropriate rescalings

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

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

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

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

a fredholm determinant identity and the convergence of moments for random young tableaux we obtain an identity between fredholm determinants of two kinds of operators one acting on functions on the unit circle and the other acting on functions on a subset of the integers this identity is a generalization of an identity between a toeplitz determinant and a fredholm determinant that has appeared in the random permutation context using this identity we prove in particular convergence of moments for arbitrary rows of a random young diagram under plancherel measure

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

convergence analysis of the defect correction iteration for hyperbolic problems this paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady euler equations such equations are also commonly solved by pseudo time integration the steady solution being achieved as the limit for t to infty of the solution of a time dependent problem implicit schemes are then often chosen for their favourable stability properties permitting large timesteps for efficiency an important class of implicit schemes involving first and second order accurate upwind discretisations is considered in the limit of an infinite timestep these implicit schemes approach defect correction algorithms thus our analysis is informative for both types of construction simple scalar linear model problems are introduced for the one dimensional and two dimensional cases these model problems are analyzed in detail by both fourier and matrix analyses the convergence behaviour appears to be strongly dependent on a parameter beta

multigrid and defect correction for the steady navier stokes equations theoretical and experimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full steady 2d compressible navier stokes equations iterative defect correction is introduced for circumventing the difficulty in solving navier stokes equations discretized with a second or higher order accurate convective part by fourier analysis applied to a model equation an optimal choice is made for the operator to be inverted in the defect correction iteration as a smoothing technique for the multigrid method collective symmetric point gauss seidel relaxation is applied with as the basic solution technique exact newton iteration applied to a continuously differentiable first order upwind discretization of the full navier stokes equations for non smooth flow problems the convergence results obtained are already competitive with those of well established navier stokes methods for smooth flow problems the present method performs better than any standard method here first order discretization error accuracy is attained in a single multigrid cycle and second order accuracy in only one defect correction cycle the method contributes to the state of the art in efficiently computing compressible viscous flows

bethe ansatz solution of a closed spin 1 xxz heisenberg chain with quantum algebra symmetry a quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail the bethe ansatz equations as well as the energy eigenvalues of the model are obtained the highest weight property of the bethe vectors with respect to uq sl 2 is proved

multigrid iterative algorithm using pseudo compressibility for three dimensional mantle convection with strongly variable viscosity a numerical algorithm for solving mantle convection problems with strongly variable viscosity is presented equations for conservation of mass and momentum for highly viscous and incompressible fluids are solved iteratively by a multigrid method in combination with pseudo compressibility and local time stepping techniques this algorithm is suitable for large scale three dimensional numerical simulations because i memory storage for any additional matrix is not required and ii vectorization and parallelization are straightforward the present algorithm has been incorporated into a mantle convection simulation program based on the finite volume discretization in a three dimensional rectangular domain benchmark comparisons with previous two and three dimensional calculations including the temperature and or depth dependent viscosity revealed that accurate results are successfully reproduced even for the cases with viscosity variations of several orders of magnitude the robustness of the numerical method against viscosity variation can be significantly improved by increasing the pre and post smoothing calculations during the multigrid operations and the convergence can be achieved for the global viscosity variations up to 10 1 0

nonintersecting brownian bridges on the unit circle with drift abstract nonintersecting brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers we investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position for each return time t 2 we show there exists a critical drift c t such that if 2 m t c t for some integer m then the expected winding number for each walker is asymptotically m in addition we compute the asymptotic distribution of total winding numbers in the double scaling regime in which the expected number of walkers with winding number not equal to m is finite the method of proof is riemann hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights this is the first asymptotic analysis of such a class of polynomials we determine asymptotic formulas for these polynomials as the degree of the polynomial grows large and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights

completeness of bethe s states for the generalized xxz model we study the bethe ansatz equations for a generalized xxz model on a one dimensional lattice assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of bethe s states for a generalized xxz model we find an exact form for the inverse matrix related with vacancy numbers and compute its determinant this inverse matrix has a tridiagonal form generalizing the cartan matrix of type a

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices