a level set approach for simulations of flows with multiple moving contact lines with hysteresis the level set method of sussman et al j comput phys 148 1999 81 is extended such that flows with multiple moving contact lines can be simulated accounting for inertia a relation between contact line speed and contact angle slip and contact line hysteresis the convergence properties of the method are investigated with particular attention on the resolution of the contact line stress singularity results are compared with a lubrication theory for spreading droplets

asymptotics of a fredholm determinant corresponding to the first bulk critical universality class in random matrix models we study the determinant det i kpii of an integrable fredholm operator kpii acting on the interval s s whose kernel is constructed out of the function associated with the hastings mcleod solution of the second painleve equation this fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density using the riemann hilbert method we evaluate the large s asymptotics of det i kpii

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

the asymptotics of a bessel kernel determinant which arises in random matrix theory in random matrix theory the local correlations of the laguerre and jacobi unitary ensemble in the hard edge scaling limit can be described in terms of the bessel kernel b x y xyj x yj y j y xj x x2 y2 x y 0 1 in particular the so called hard edge gap probabilities p r can be expressed as the fredholm determinants of the corresponding integral operator b restricted to the finite interval 0 r using operator theoretic methods we are going to compute their asymptotics as r i e we show that p r det i b l2 0 r exp r24 r 22logr g 1 2 2 where g stands for the barnes g function in fact this asymptotic formula will be proved for all complex parameters satisfying re 1

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

limiting distribution of maximal crossing and nesting of poissonized random matchings the notion of r crossing and r nesting of a complete matching was introduced and a symmetry property was proved by chen et al trans amer math soc 359 2007 1555 1575 we consider random matchings of large size and study their maximal crossing and their maximal nesting it is known that the marginal distribution of each of them converges to the goe tracy widom distribution we show that the maximal crossing and the maximal nesting becomes independent asymptotically and we evaluate the joint distribution for the poissonized random matchings explicitly to the first correction term this leads to an evaluation of the asymptotic of the covariance furthermore we compute the explicit second correction term in the distribution function of two objects a the length of the longest increasing subsequence of poissonized random permutation and b the maximal crossing and hence also the maximal nesting of poissonized random matching

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

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

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

recursion for the smallest eigenvalue density of β wishart laguerre ensemble the statistics of the smallest eigenvalue of wishart laguerre ensemble is important from several perspectives the smallest eigenvalue density is typically expressible in terms of determinants or pfaffians these results are of utmost significance in understanding the spectral behavior of wishart laguerre ensembles and among other things unveil the underlying universality aspects in the asymptotic limits however obtaining exact and explicit expressions by expanding determinants or pfaffians becomes impractical if large dimension matrices are involved for the real matrices beta 1 edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices very recently an analogous recurrence scheme has been obtained for the complex matrices beta 2 in the present work we extend this to beta wishart laguerre ensembles for the case when exponent alpha in the associated laguerre weight function lambda alpha e beta lambda 2 is a non negative integer while beta is positive real this also gives access to the smallest eigenvalue density of fixed trace beta wishart laguerre ensemble as well as moments for both cases moreover comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly exact evaluations for large values of n the matrix dimension and alpha also enable us to compare with tracy widom density and large deviation results of katzav and castillo we also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the wigner smith matrix and are relevant to the problem of quantum chaotic scattering

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

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

local behavior and hitting probabilities of the text airy _1 process we obtain a formula for the n dimensional distributions of the text airy 1 process in terms of a fredholm determinant on l 2 mathbb r as opposed to the standard formula which involves extended kernels on l 2 1 dots n times mathbb r the formula is analogous to an earlier formula of prahofer and spohn j stat phys 108 5 6 1071 1106 2002 for the text airy 2 process using this formula we are able to prove that the text airy 1 process is holder continuous with exponent frac 1 2 and that it fluctuates locally like a brownian motion we also explain how the same methods can be used to obtain the analogous results for the text airy 2 process as a consequence of these two results we derive a formula for the continuum statistics of the text airy 1 process analogous to that obtained in corwin et al commun math phys 2011 to appear for the text airy 2 process

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

compensated compactness for differential forms in carnot groups and applications abstract in this paper we prove a compensated compactness theorem for differential forms of the intrinsic complex of a carnot group the proof relies on an l s hodge decomposition for these forms because of the lack of homogeneity of the intrinsic exterior differential hodge decomposition is proved using the parametrix of a suitable 0 order laplacian on forms

edge statistics for a class of repulsive particle systems we study a class of interacting particle systems on mathbb r which was recently investigated by gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 these ensembles generalize eigenvalue ensembles of hermitian random matrices by allowing different interactions between particles although these ensembles are not known to be determinantal one can use the stochastic linearization method of gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 to represent them as averages of determinantal ones our results describe the transition between universal behavior in the regime of the tracy widom law and non universal behavior for large deviations of the rightmost particle moreover a detailed analysis of the transition that occurs in the regime of moderate deviations is provided we also compare our results with the corresponding ones obtained recently for determinantal ensembles eichelsbacher et al in symmetry integr geom methods appl 12 18 2016 doi 10 3842 sigma 2016 093 schuler in moderate large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles ph d thesis university of bayreuth 2015 in particular we discuss how the averaging effects the leading order behavior in the regime of large deviations in the analysis of the averaging procedure we use detailed asymptotic information on the behavior of christoffel darboux kernels that is uniform for perturbative families of weights such results have been provided by k schubert k schuler and the authors in kriecherbauer et al markov process relat fields 21 3 part 2 639 694 2015

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

the widom dyson constant for the gap probability in random matrix theory in the bulk scaling limit for the gaussian unitary ensemble in random matrix theory the probability that there are no eigenvalues in the interval 0 2s is given by p s det i k s where k s is the trace class operator with kernel k s x y sin x y p x y acting on l 2 0 2s in the analysis of the asymptotic behavior of p s as s there is particular interest in the constant term known as the widom dyson constant we present a new derivation of this constant which can be adapted to calculate similar critical constants in other problems arising in random matrix theory

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

on a class of semilinear evolution equations for vector potentials associated with maxwell s equations in carnot groups abstract in this paper we prove existence and regularity results for a class of semilinear evolution equations that are satisfied by vector potentials associated with maxwell s equations in carnot groups connected simply connected stratified nilpotent lie groups the natural setting for these equations is provided by the so called rumin s complex of intrinsic differential forms

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

simulation of moving contact lines in two phase polymeric fluids we numerically study the influence of polymer additives on contact line dynamics using a sharp interface model an additional term which accounts for the polymer stress is added to the two phase navier stokes equation the polymer stress is modeled by the fene p model the coupled dynamic equations are solved using the finite difference method the interface is tracked using marker particles and the interface conditions are imposed using the immersed boundary method simulations in a couette geometry are carried out to study the effect of the polymer stress on the moving contact line it is found that the large velocity gradient near the moving contact line results in large polymer stress which retards the contact line motion the polymer stress increases with the polymer viscosity but is insensitive to the change of the weissenberg number the strength of the polymer stress is also found to increase with the capillary number especially when the capillary number is large we also study the relation between the apparent contact angle and the capillary number for newtonian fluids the relation fits well with cox s analytic result however deviation occurs for polymeric fluids the critical capillary number at which the transition to an unsteady state happens decreases as the polymer viscosity increases

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

from gap probabilities in random matrix theory to eigenvalue expansions we present a method to derive asymptotics of eigenvalues for trace class integral operators acting on a single interval which belongs to the ring of integrable operators its et al 1990 int j mod phys b 4 1003 37 our emphasis lies on the behavior of the spectrum of k as and i is fixed we show that this behavior is intimately linked to the analysis of the fredholm determinant as and in a stokes type scaling regime concrete asymptotic formulae are obtained for the eigenvalues of airy and bessel kernels in random matrix theory

universal fluctuations in kardar parisi zhang growth on one dimensional flat substrates we present a numerical study of the evolution of height distributions hds obtained in interface growth models belonging to the kardar parisi zhang kpz universality class the growth is done on an initially flat substrate the hds obtained for all investigated models are very well fitted by the theoretically predicted gaussian orthogonal ensemble goe distribution the first cumulant has a shift that vanishes as t 1 3 while the cumulants of order 2 n 4 converge to goe as t 2 3 or faster behaviors previously observed in other kpz systems these results yield evidences for the universality of the goe distribution in kpz growth on flat substrates finally we further show that the surfaces are described by the airy 1 process

exact solvability and asymptotic aspects of generalized xx0 spin chains abstract building on our earlier work saeedian et al 2018 we introduce and study generalized xx0 models we explicitly construct a long range interacting spin chain referred to as the selberg model and study the correlation functions of the selberg and xx0 models using a matrix integral representation of the generalized xx0 model and applying asymptotic analysis in non intersecting brownian motion the phase structure of the selberg model is determined we find that tails of the tracy widom distribution of gaussian unitary ensemble govern a discrete to continuous third order phase transition in selberg model the same method also reproduces the gross witten phase transition of the original xx0 model finally we conjecture universal features for the phase structure of the generalized xx0 model

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

3d phase field simulations of interfacial dynamics in newtonian and viscoelastic fluids this work presents a three dimensional finite element algorithm based on the phase field model for computing interfacial flows of newtonian and complex fluids a 3d adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk it is key to accurate resolution of the interface at manageable computational costs the coupled navier stokes and cahn hilliard equations plus the constitutive equation for non newtonian fluids are solved using second order implicit time stepping within each time step newton iteration is used to handle the nonlinearity and the linear algebraic system is solved by preconditioned krylov methods the phase field model with a physically diffuse interface affords the method several advantages in computing interfacial dynamics one is the ease in simulating topological changes such as interfacial rupture and coalescence another is the capability of computing contact line motion without invoking ad hoc slip conditions as validation of the 3d numerical scheme we have computed drop deformation in an elongational flow relaxation of a deformed drop to the spherical shape and drop spreading on a partially wetting substrate the results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate excellent agreement is achieved provided that the 3d interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid since our model involves several coupled partial differential equations and we use a fully implicit scheme the matrix inversion requires a large memory this puts a limit on the scale of problems that can be simulated in 3d especially for viscoelastic fluids

dyson s constants in the asymptotics of the determinants of wiener hopf hankel operators with the sine kernel let k alpha pm stand for the integral operators with the sine kernels frac sin x y pi x y pm frac sin x y pi x y acting on l2 0 dyson conjectured that the asymptotics of the fredholm determinants of i k alpha pm are given by log det i k alpha pm frac alpha 2 4 mp frac alpha 2 frac log alpha 8 frac log 2 24 pm frac log 2 4 frac 3 2 zeta 1 o 1 as in this paper we are going to give a proof of these two asymptotic formulas

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

monodromy dependence and connection formulae for isomonodromic tau functions we discuss an extension of the jimbo miwa ueno differential 1 form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients this extension is based on the results of m bertola generalizing a previous construction by b malgrange we show how this 1 form can be used to solve a long standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors we explain how this scheme works for fuchsian systems and in particular calculate the connection constant for the generic painleve vi tau function the result proves the conjectural formula for this constant proposed by iorgov lisovyy and tykhyy we also apply the method to non fuchsian systems and evaluate constant factors in the asymptotics of the painleve ii tau function

on integrals of the tronquee solutions and the associated hamiltonians for the painleve ii equation abstract we consider a family of tronquee solutions of the painleve ii equation q s 2 q s 3 s q s 2 1 2 1 2 which is characterized by the stokes multipliers s 1 e 2 i s 2 s 3 e 2 i with being a free parameter these solutions include the well known generalized hastings mcleod solution as a special case if 0 we derive asymptotics of integrals of the tronquee solutions and the associated hamiltonians over the real axis for 1 2 and 0 with the constant terms evaluated explicitly our results agree with those already known in the literature if the parameters and are chosen to be special values some applications of our results in random matrix theory are also discussed

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

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

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

multiple gamma and related functions the authors give several new and potentially useful relationships between the multiple gamma functions and other mathematical functions and constants as by products of some of these relationships a classical definite integral due to euler and other definite integrals are also considered together with closed form evaluations of some series involving the riemann and hurwitz zeta functions

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process