superconvergent explicit two step peer methods we consider explicit two step peer methods for the solution of nonstiff differential systems by an additional condition a subclass of optimally zero stable methods is identified that is superconvergent of order p s 1 where s is the number of stages the new condition allows us to reduce the number of coefficients in a numerical search for good methods we present methods with 4 7 stages which are tested in fortran90 and compared with dopri5 and dop853 the results confirm the high potential of the new class of methods

general β jacobi corners process and the gaussian free field we prove that the two dimensional gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general jacobi random matrix ensembles our approach is based on the connection of the jacobi ensembles to a degeneration of the macdonald processes that parallels the degeneration of the macdonald polynomials to the heckman opdam hypergeometric functions of type a we also discuss the limit 2015 wiley periodicals inc

nonintersecting random walks in the neighborhood of a symmetric tacnode consider a continuous time random walk in z with independent and exponentially distributed jumps 1 the model in this paper consists in an infinite number of such random walks starting from the complement of m m 1 m 1 m at time t returning to the same starting positions at time t and conditioned not to intersect this yields a determinantal process whose gap probabilities are given by the fredholm determinant of a kernel thus this model consists of two groups of random walks which are contained within two ellipses which with the choice m 2t to leading order just touch so we have a tacnode we determine the new limit extended kernel under the scaling m 2t t1 3 where parameter controls the strength of interaction between the two groups of random walkers

adaptive ode solvers in extended kalman filtering algorithms this paper studies numerical methods that are used to treat ordinary differential equations arising in extended kalman filtering algorithms such problems are called the moment equations and constitute an important subclass of differential equations they all possess a number of specific properties that make the standard available software ineffective for some mathematical models in this area for instance we show that built in matlab ode solvers are not able to cope with a number of tests considered here in contrast the hybrid method developed by mazzoni 2007 exhibits much better performance on the same examples the latter means that specially designed numerical schemes are needed for efficient solution of the moment equations instead of the standard software designed for general systems of ordinary differential equations

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

approximations for expected generation number a deterministic formula is commonly used to approximate the expected generation number of a population of growing cells however this can give misleading results because it does not allow for natural variation in the times that individual cells take to reproduce here we present more accurate approximations for both symmetric and asymmetric cell division based on the first two moments of the generation time distribution these approximations are also robust we illustrate the improved approximations using data that arise from monitoring individual yeast cells under a microscope and also demonstrate how the approximations can be used when such detailed data are not available

dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition the one dimensional totally asymmetric simple exclusion process tasep is considered we study the time evolution property of a tagged particle in the tasep with the step initial condition calculated is the multi time joint distribution function of its position using the relation of the dynamics of the tasep to the schur process we show that the function is represented as the fredholm determinant we also study the scaling limit the universality of the largest eigenvalue in the random matrix theory is realized in the limit when the hopping rates of all particles are the same it is found that the joint distribution function converges to that of the airy process after the time at which the particle begins to move on the other hand when there are several particles with small hopping rate in front of a tagged particle the limiting process changes at a certain time from the airy process to the process of the largest eigenvalue in the hermitian multi matrix model with external sources

geometric rsk correspondence whittaker functions and symmetrized random polymers we show that the geometric lifting of the rsk correspondence introduced by a n kirillov physics and combinatorics proc nagoya 2000 2nd internat workshop pp 82 150 2001 is volume preserving with respect to a natural product measure on its domain and that the integrand in givental s integral formula for mathit gl n mathbb r whittaker functions arises naturally in this context apart from providing further evidence that whittaker functions are the natural analogue of schur polynomials in this setting our results also provide a new combinatorial framework for the study of random polymers when the input matrix consists of random inverse gamma distributed weights the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of whittaker functions next we restrict the geometric rsk mapping to symmetric matrices and show that the volume preserving property continues to hold we determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal with an additional factor on the main diagonal the third combinatorial mapping studied is a variant of the geometric rsk mapping for triangular arrays which is again showed to be volume preserving this leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal we also show that the analogues of the cauchy littlewood identity in the setting of this paper are equivalent to a collection of whittaker integral identities conjectured by bump number theory trace formulas and discrete groups pp 49 109 1989 and bump and friedberg festschrift in honor of piatetski shapiro part ii pp 47 65 1990 and proved by stade am j math 123 121 161 2001 israel j math 127 201 219 2002 our approach leads to new combinatorial proofs and generalizations of these identities with some restrictions on the parameters

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

airy processes with wanderers and new universality classes consider n m nonintersecting brownian bridges with n of them leaving from 0 at time t 1 and returning to 0 at time t 1 while the m remaining ones wanderers go from m points a i to m points b i first we keep m fixed and we scale a i b i appropriately with n in the large n limit we obtain a new airy process with wanderers in the neighborhood of root 2n the approximate location of the rightmost particle in the absence of wanderers this new process is governed by an airy type kernel with a rational perturbation letting the number m of wanderers tend to infinity as well leads to two pearcey processes about two cusps a closing and an opening cusp the location of the tips being related by an elliptic curve upon tuning the starting and target points one can let the two tips of the cusps grow very close this leads to a new process which might be governed by a kernel represented as a double integral involving the exponential of a quintic polynomial in the integration variables

generalizing global error estimation for ordinary differential equations by using coupled time stepping methods abstract this study introduces new time stepping strategies with built in global error estimators the new methods propagate the defect along with the numerical solution much like solving for the correction or zadunaisky s procedure however the proposed approach allows for overlapped internal computations and therefore represents a generalization of the classical numerical schemes for solving differential equations with global error estimation the resulting algorithms can be effectively represented as general linear methods several explicit self starting schemes akin to runge kutta methods with global error estimation are introduced and the theoretical considerations are illustrated in several examples

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

generalized green functions and current correlations in the tasep we study correlation functions of the totally asymmetric simple exclusion process tasep in discrete time with backward sequential update we prove a determinantal formula for the generalized green function which describes transitions between positions of particles at different individual time moments in particular the generalized green function defines a probability measure at staircase lines on the space time plane the marginals of this measure are the tasep correlation functions in the space time region not covered by the standard green function approach as an example we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance an asymptotic analysis shows that current fluctuations converge to the airy2 process

log gamma polymer free energy fluctuations via a fredholm determinant identity we prove that under n1 3 scaling the limiting distribution as n of the free energy of seppalainen s log gamma discrete directed polymer is gue tracy widom the main technical innovation we provide is a general identity between a class of n fold contour integrals and a class of fredholm determinants applying this identity to the integral formula proved in corwin et al tropical combinatorics and whittaker functions http arxiv org abs 1110 3489v3 math pr 2012 for the laplace transform of the log gamma polymer partition function we arrive at a fredholm determinant which lends itself to asymptotic analysis and thus yields the free energy limit theorem the fredholm determinant was anticipated in borodin and corwin macdonald processes http arxiv org abs 1111 4408v3 math pr 2012 via the formalism of macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach

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

thermodynamic limit of random partitions and dispersionless toda hierarchy we study the thermodynamic limit of random partition models for the instanton sum of 4d and 5d supersymmetric u 1 gauge theories deformed by some physical observables the physical observables correspond to external potentials in the statistical model the partition function is reformulated in terms of the density function of maya diagrams the thermodynamic limit is governed by a limit shape of young diagrams associated with dominant terms in the partition function the limit shape is characterized by a variational problem which is further converted to a scalar valued riemann hilbert problem this riemann hilbert problem is solved with the aid of a complex curve which may be thought of as the seiberg witten curve of the deformed u 1 gauge theory this solution of the riemann hilbert problem is identified with a special solution of the dispersionless toda hierarchy that satisfies a pair of generalized string equations the generalized string equations for the 5d gauge theory are shown to be related to hidden symmetries of the statistical model the prepotential and the seiberg witten differential are also considered

parallel two step w methods with peer variables a new class of methods for the solution of stiff initial value problems is introduced that is parallel by design it has a two step character and propagates s different peer solution variables with essentially identical characteristics from step to step the main work lies in the solution of s independent linear stage equations which may be solved in parallel convergence of order s 1 and stability for general stepsize sequences are proved conditions for order s and stronger stability criteria are addressed as well promising methods up to order 7 are identified by numerical tests with some widely used stiff test problems some of these are competitive with existing software even in sequential computations

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

tacnode gue minor processes and double aztec diamonds we study determinantal point processes arising in random domino tilings of a double aztec diamond a region consisting of two overlapping aztec diamonds at a turning point in a single aztec diamond where the disordered region touches the boundary the natural limiting process is the gue minor process increasing the size of a double aztec diamond while keeping the overlap between the two aztec diamonds finite we obtain a new determinantal point process which we call the tacnode gue minor process this process can be thought of as two colliding gue minor processes as part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode gue minor process we find the inverse kasteleyn matrix for the dimer model version of the double aztec diamond

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

saddlepoint approximations to the moments of multitype age dependent branching processes with applications this article proposes saddlepoint approximations to the expectation and variance covariance function of multitype age dependent branching processes the proposed approximations are found accurate easy to implement and much faster to compute than by simulating the process multiple applications are presented including the analyses of clonal data on the generation of oligodendrocytes from their immediate progenitor cells and on the proliferation of hela cells new estimators are also constructed to analyze clonal data the proposed methods are finally used to approximate the distribution of the generation which has recently found several applications in cell biology

quasi likelihood and its application a general approach to optimal parameter estimation the general framework an alternative approach e sufficiency asymptotic confidence zones of minimum size asymptotic quasi likelihood combining estimating functions projected quasi likelihood bypassing the likelihood hypothesis testing infinite dimensional problems miscellaneous applications consistency and asymptotic normality for estimating functions complements and strategies for application

moderate deviation and exit time estimates for stationary last passage percolation we consider planar stationary exponential last passage percolation in the positive quadrant with boundary weights for rho in 0 1 and points v n 1 rho 2 n rho 2 n going to infinity along the characteristic direction we establish right tail estimates with the optimal exponent for the exit time of the geodesic along with optimal exponent estimates for the upper tail moderate deviations for the passage time for the case rho frac 1 2 in the stationary model we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time our arguments are based on moderate deviation estimates for point to point and point to line exponential last passage percolation which are obtained via random matrix estimates

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

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

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

doubly quasi consistent parallel explicit peer methods with built in global error estimation recently kulikov presented the idea of double quasi consistency which facilitates global error estimation and control considerably more precisely a local error control implemented in such methods plays a part of global error control at the same time however kulikov studied only nordsieck formulas and proved that there exists no doubly quasi consistent scheme among those methods here we prove that the class of doubly quasi consistent formulas is not empty and present the first example of such sort this scheme belongs to the family of superconvergent explicit two step peer methods constructed by weiner schmitt podhaisky and jebens we present a sample of s stage doubly quasi consistent parallel explicit peer methods of order s 1 when s 3 the notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi consistent peer method and hence its global error at the same time numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi consistent numerical integration techniques is capable of producing numerical solutions for user supplied accuracy conditions in automatic mode

global error estimation based on the tolerance proportionality for some adaptive runge kutta codes modern codes for the numerical solution of initial value problems ivps in odes are based in adaptive methods that for a user supplied tolerance d attempt to advance the integration selecting the size of each step so that some measure of the local error is email protected although this policy does not ensure that the global errors are under the prescribed tolerance after the early studies of stetter considerations concerning a theory for ode solvers in r burlisch r d grigorieff j schroder eds numerical treatment of differential equations proceedings of oberwolfach 1976 lecture notes in mathematics vol 631 springer berlin 1978 pp 188 200 tolerance proportionality in ode codes in r marz ed proceedings of the second conference on numerical treatment of ordinary differential equations humbold university berlin 1980 pp 109 123 and the extensions of higham global error versus tolerance for explicit runge kutta methods ima j numer anal 11 1991 457 480 the tolerance proportionality of adaptive ode solvers j comput appl math 45 1993 227 236 the reliability of standard local error control algorithms for initial value ordinary differential equations in proceedings the quality of numerical software assessment and enhancement ifip series springer berlin 1997 it has been proved that in many existing explicit runge kutta codes the global errors behave asymptotically as some rational power of d this step size policy for a given ivp determines at each grid point t n a new step size h n 1 h t n d so that h t d is a continuous function of t in this paper a study of the tolerance proportionality property under a discontinuous step size policy that does not allow to change the size of the step if the step size ratio between two consecutive steps is close to unity is carried out this theory is applied to obtain global error estimations in a few problems that have been solved with the code gauss2 s gonzalez pinto r rojas bello gauss2 a fortran 90 code for second order initial value problems based on an adaptive two stage runge kutta gauss method with this discontinuous step size policy

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

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

the tolerance proportionality of adaptive ode solvers abstract modern software for solving ordinary differential equation ode initial value problems requires the user to specify the ode and choose a value for the error tolerance the software can be thought of as a black box with a dial turning the dial changes the accuracy and expense of the integration process it is therefore of interest to know how the global error varies with the error tolerance in this work we look at explicit runge kutta methods and show that with any standard error control method and ignoring higher order terms the global error in the numerical solution behaves like a known rational power of the error tolerance this generalises earlier work of stetter who found sufficient conditions for the global error to be linear in the tolerance we also display the order of the next highest term we then analyse continuous runge kutta schemes and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula finally we extend the results to the case of ode systems with constant delays thereby generalising some previous results of the author

new third and fourth order singly diagonally implicit two step peer triples with local and global error controls for solving stiff ordinary differential equations in this paper we present new singly diagonally implicit two step peer triples equipped with local and global error controls for providing preassigned accuracies of numerical integration of stiff ordinary differential equations odes in automatic mode recently kulikov and weiner 2015 reported an efficient numerical integration tool of order 2 which solves accurately many difficult stiff odes including large scale systems obtained from semidiscretization of partial differential equations pdes corresponding to user supplied requests moreover the cited method is not only accurate but it is also more efficient than the well known matlab code ode23s with local error control here we further extend the published technique and construct variable stepsize singly diagonally implicit two step peer triples of the higher orders 3 and 4 our numerical experiments suggest that these triples are suitable for treating stiff problems with prescribed accuracy conditions in addition performance of the presented methods can be comparable to that of the built in stiff matlab code ode15s with local error control which is considered to be a benchmark means for solving stiff odes by many practitioners for some test problems

new approximations to the malthusian parameter approximations to the malthusian parameter of an age dependent branching process are obtained in terms of the moments of the lifetime distribution by exploiting a link with renewal theory in several examples the new approximations are more accurate than those currently in use even when based on only the first two moments the new approximations are extended to include a form of asymmetric cell division that occurs in some species of yeast when used for inference the new approximations are shown to have high efficiency

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

tilings of non convex polygons skew young tableaux and determinantal processes this paper studies random lozenge tilings of general non convex polygonal regions we show that the pairwise interaction of the non convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations the precise geometrical figure here consists of a hexagon with cuts along opposite edges for this model we take limits when the size of the hexagon and the cuts tend to infinity while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit we show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral where the number of integrations is related to the fixed geometric data above the limiting kernel is believed to be a universal master kernel

directed last passage percolation and random matrices this paper is a survey of various techniques recent results and applications of directed last passage percolation model and its relation to random matrices the connection can be explained mainly via the universality of famous tracy widom distribution and airy process some unsolved problems are also surveyed

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

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

global error estimation and the backward differentiation formulas the way in which global error estimates might be used to control global error is discussed various global error estimation techniques are surveyed with special attention to defect correction in the discrete realm or difference correction and to those techniques that use a continuous approximation to the solution the problems of stiffness and rapidly varying stepsize is discussed for global error estimation for the bdfs and an unusual solution proposed

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

on global error estimation and control for initial value problems this paper addresses global error estimation and control for initial value problems for ordinary differential equations the focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation control is achieved through tolerance proportionality both approaches are found to work well and to enable estimation and control in a reliable manner

partially implicit peer methods for the compressible euler equations when cut cells are used for the representation of orography in numerical weather prediction models this leads to very small cells on one hand this results in very harsh time step restrictions for explicit methods due to the cfl criterion on the other hand cut cells only appear in a small region of the domain therefore we consider a partially implicit method in cut cells the jacobian incorporates advection diffusion and acoustics while in the full cells of the free atmosphere only the acoustic part is used i e the method is linearly implicit in the cut cell regions and semi explicit in the free regions and computes with time step sizes restricted only by the cfl condition in the free atmosphere furthermore we use a simplified jacobian in the cut cell regions in order to save storage and gain computational efficiency while the method retains the order independently of the jacobian we present a linear stability theory which takes the effects of the simplifications of the jacobian on stability into account the presented method is as stable and accurate as the underlying split explicit method but furthermore it can compute with cut cells with nearly no additional effort

on quasi consistent integration by nordsieck methods in this paper we study advantages of numerical integration by quasi consistent nordsieck formulas all quasi consistent numerical methods possess at least one important property for practical use which has not attracted attention yet i e the global error of a quasi consistent method has the same order as its local error this means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms in other words the global error control can be as cheap as the local error control in the methods under discussion here we apply the above mentioned idea to nordsieck adams moulton methods which are known to be quasi consistent moreover some nordsieck adams moulton methods are even super quasi consistent the latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods in turn this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control we also introduce implicitly extended nordsieck methods such that in some sense they form pairs of embedded formulas with their source nordsieck methods this facilitates the local error control in quasi consistent nordsieck schemes numerical examples presented in this paper confirm clearly the power of quasi consistent integration in practice

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

airy kernel with two sets of parameters in directed percolation and random matrix theory we introduce a generalization of the extended airy kernel with two sets of real parameters we show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

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

from aztec diamonds to pyramids steep tilings we introduce a family of domino tilings that includes tilings of the aztec diamond and pyramid partitions as special cases these tilings live in a strip of mathbb z 2 of the form 1 leq x y leq 2 ell for some integer ell geq 1 and are parametrized by a binary word w in 2 ell that encodes some periodicity conditions at infinity aztec diamond and pyramid partitions correspond respectively to w ell and to the limit case w infty infty for each word w and for different types of boundary conditions we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips that admits a natural multivariate generalization the main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism which we slightly extend in order to handle littlewood type identities in probabilistic terms our tilings map to schur processes of different types standard pfaffian and periodic we also introduce a more general model that interpolates between domino tilings and plane partitions

doubly quasi consistent fixed stepsize numerical integration of stiff ordinary differential equations with implicit two step peer methods abstract the problem of accurate and efficient numerical integration of ordinary differential equations odes is a key issue of computation science which has a valued practical impact in various topics of research in this respect effective global error estimation and control strategies are of particular theoretical and practical significance they allow many applied dynamic systems to be treated numerically for user supplied accuracy conditions in automatic mode the property of double quasi consistency plays an outstanding role in the mentioned topic because the global or true error of a doubly quasi consistent method is asymptotically equal to its local error therefore this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run the latter was considered to be impossible for decades the property of double quasi consistency goes back to 2009 when kulikov studied it in the first time but failed to find doubly quasi consistent numerical methods among nordsieck formulas later on kulikov and weiner proved the existence of doubly quasi consistent numerical schemes among superconvergent explicit two step parallel peer methods and examined their practical efficiency in comparison to explicit matlab ode solvers in 2010 the focus of the present research is on accurate numerical integration formulas for treating stiff odes which often arise in practice and for which explicit methods are shown to be ineffective here we make the first step towards an accurate and efficient numerical solution of stiff odes and prove existence of implicit doubly quasi consistent formulas we fulfill our investigation of double quasi consistency within the family of fixed stepsize implicit two step peer schemes and construct two methods of convergence orders 3 and 4 which possess excellent stability properties our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

point to line last passage percolation and the invariant measure of a system of reflecting brownian motions this paper proves an equality in law between the invariant measure of a reflected system of brownian motions and a vector of point to line last passage percolation times in a discrete random environment a consequence describes the distribution of the all time supremum of dyson brownian motion with drift a finite temperature version relates the point to line partition functions of two directed polymers with an inverse gamma and a brownian environment and generalises dufresne s identity our proof introduces an interacting system of brownian motions with an invariant measure given by a field of point to line log partition functions for the log gamma polymer

a mixture model with dependent observations for the analysis of csfe labeling experiments recent advances in flow cytometry have resulted in the development of powerful bioassays to analyze the proliferation of cell populations the carboxy fluorescein diacetate succinimidyl ester cfse labeling experiment is one such assay that is widely used to study cell kinetics and as a standard tool for investigating lymphocyte proliferation several mathematical models have been proposed to describe cell proliferation during cfse experiments the statistical analysis of cfse labeling data has received little attention but it poses a number of methodological issues in this article we approach their analysis using a mixture model the mixing proportions are specified through an age dependent branching process that models the temporal organization of the cell population because the cfse molecules are partitioned between daughter cells when their mother divides the observations generated by this assay are dependent the data structure can be compared with that of a partially observed random effects mod

analysis of error control strategies for continuous runge kutta methods there has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete runge kutta methods with the availability of these schemes it can now be asked that a numerical method provide a continuous approximation to the solution thispaper rather than view such a continuous method as an interpolant superimposed on a standard discrete method considers how the interpolant and its associated defect can be effectively used in the underlying error and stepsize control mechanism in particular four error control strategies are considered that can be used in methods based on runge kutta formula pairs with interpolants an asymptotic and a nonasymptotic analysis of each strategy are presented it is shown that a strategy based on direct defect control can provide significant advantages over existing strategies with only a modest increase in cost

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

limit processes for tasep with shocks and rarefaction fans we consider the totally asymmetric simple exclusion process tasep with two sided bernoulli initial condition i e with left density and right density we study the associated height function whose discrete gradient is given by the particle occurrences macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of we characterize the large time scaling limit of the multipoint fluctuations as a function of the densities and of the different macroscopic regions moreover using a slow decorrelation phenomena the results are extended from fixed time to the whole space time except along the some directions the characteristic solutions of the related burgers equation where the problem is still open

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

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

global error estimation with runge kutta triples abstract the applications of runge kutta rk interpolation to global error estimation using the zadunaisky and related techniques are considered it is shown that the pseudo problem can be based on dense output values within any one step and reliable global error estimates can be obtained at the integration mesh points by using special rk formulae some special formulae of orders 2 6 are presented together with numerical results

variable stepsize interpolating explicit parallel peer methods with inherent global error control in this paper we present variable stepsize explicit parallel peer methods grounded in the interpolation idea approximation stability and convergence are studied in detail in particular we prove that some interpolating peer methods are stable on any variable mesh in practice double quasi consistency is utilized to introduce an efficient global error estimation formula in the numerical methods under discussion the main advantage of these new adaptive schemes is the fact that the leading terms of their local and true errors coincide thus controlling the local error of such methods by cheap standard techniques automatically regulates their global error as well numerical experiments of this paper support theoretical results presented below and illustrate how the new global error control concept works in practice we also conduct a comparison with explicit ode solvers in matlab

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

limit process of stationary tasep near the characteristic line the totally asymmetric simple exclusion process tasep on input amssym bbb z with the bernoulli measure as an initial condition 0 1 is stationary it is known that along the characteristic line the current fluctuates at an order of t1 3 the limiting distribution has also been obtained explicitly in this paper we determine the limiting multipoint distribution of the current fluctuations moving away from the characteristics by the order t2 3 the main tool is the analysis of a related directed last percolation model we also discuss the process limit in tandem queues in equilibrium 2010 wiley periodicals inc

global error estimation and control in linearly implicit parallel two step peer w methods the class of linearly implicit parallel two step peer w methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations those schemes allow for parallelism across the method that is an important feature for implementation on modern computational devices most importantly all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration this property results in the fact that no order reduction occurs when they are applied to very stiff problems in this paper we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user supplied accuracy requirement in automatic mode an algorithm of such global error control and other technical particulars are also discussed here numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions including nonstiff stiff very stiff and large scale differential equations a comparison with the well known stiff solver rodas is also shown

local and global error estimation and control within explicit two step peer triples this paper elaborates a global error estimation and control mechanism in explicit two step peer methods these recently designed methods exhibit their high efficiency even in comparison to the best explicit runge kutta pairs more precisely we form here triples of the so called superconvergent explicit peer schemes and show that they are cheap and able to achieve preassigned accuracy conditions in automatic mode for comparison we present also numerical data derived by built in explicit matlab ode solvers implemented with only local error control especially we point out that a scaled global error is computed and regulated in this paper in contrast to the earlier published results where the absolute values of the global error have been utilized

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

on the principal components of sample covariance matrices we introduce a class of m times m sample covariance matrices mathcal q which subsumes and generalizes several previous models the associated population covariance matrix sigma mathbb e mathcal q is assumed to differ from the identity by a matrix of bounded rank all quantities except the rank of sigma i m may depend on m in an arbitrary fashion we investigate the principal components i e the top eigenvalues and eigenvectors of mathcal q we derive precise large deviation estimates on the generalized components langle mathbf w varvec xi i rangle of the outlier and non outlier eigenvectors varvec xi i our results also hold near the so called bbp transition where outliers are created or annihilated and for degenerate or near degenerate outliers we believe the obtained rates of convergence to be optimal in addition we derive the asymptotic distribution of the generalized components of the non outlier eigenvectors a novel observation arising from our results is that unlike the eigenvalues the eigenvectors of the principal components contain information about the subcritical spikes of sigma the proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix mathcal q satisfying mathbb e mathcal q i m as input the isotropic local marchenko pastur law established in bloemendal et al electron j probab 19 1 53 2014 level repulsion and quantum unique ergodicity of the eigenvectors the latter is a special case of a new universality result for the joint eigenvalue eigenvector distribution

global error control in adaptive nordsieck methods this paper deals with variable stepsize nordsieck formulas applied to ordinary differential equations it focuses on local and global error evaluation techniques in the mentioned numerical schemes the error estimators are derived for both consistent nordsieck methods and quasi consistent ones it is also shown how quasi consistent nordsieck formulas which suffer on variable grids from the order reduction phenomenon can be modified in an optimal way to avoid the order reduction the latter results in a new class of reducible nordsieck methods designed here in addition we supply our numerical schemes with an automatic global error control mechanism to arrive at the adaptive numerical technique for solving differential equations of preassigned quality the capacity for deriving numerical solutions satisfying user supplied accuracy conditions in automatic mode is confirmed by numerical experiments on two test problems

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

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique

numerical investigations on global error estimation for ordinary differential equations four techniques of global error estimation which are richardson extrapolation rs zadunaisky s technique zd solving for the correction sc and integration of principal error equation ipee have been compared in different integration codes dopri5 dvode dstep theoretical aspects concerning their implementations and their orders are first given second a comparison of them based on a large number of tests is presented in terms of cost and precision sc is a method of choice for one step methods it is much more precise and less costly than rs and leads to the same precision as zd for half its cost ipee can provide the order of the error for a cheap cost in codes based on one step methods in multistep codes only rs and ipee have been implemented since they are the only ones whose theoretical justification has been extended to this case there rs still provides a more reliable estimation than ipee however as these techniques are based on variations of the global error irrespective of the numerical method used they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic

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