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

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

on the dini calculus n any positive integer this is the objective of the present note the upshot being a systematic calculus for all such integrals in the even more general context of a l parameter family belonging to g r while we shall only deal explicitly with the case of the real line it would certainly be of interest to extend what is said here to the higher dimensional setting this seems to be a problem of some complexity but we shall offer a conjecture at the end of the paper on how things might be expected to develop

representations of the discrete inhomogeneous lorentz group and dirac wave equation on the lattice we propose the fundamental and two dimensional representation of the lorentz groups on a 3 1 dimensional hypercubic lattice from which representations of higher dimensions can be constructed for the unitary representation of the discrete translation group we use the kernel of the fourier transform from the dirac representation of the lorentz group including reflections we derive in a natural way the wave equation on the lattice for spin 1 2 particles finally the induced representation of the discrete inhomogeneous lorentz group is constructed by standard methods and its connection with the continuous case is discussed

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

exact results for the asymmetric simple exclusion process with a blockage we present new results for the current as a function of transmission rate in the one dimensional totally asymmetric simple exclusion process tasep with a blockage that lowers the jump rate at one site from one tor 1 exact finitevolume results serve to bound the allowed values for the current in the infinite system this proves the existence of a nonequilibrium phase transition corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state a series expansion inr derived from the finite systems is proven to be asymptotic for all sufficiently large systems pade approximants based on this series which make specific assumptions about the nature of the singularity atr 1 match numerical data for the infinite system to 1 part in 104

limiting curves for i i d records we consider the concentration of measure for n i i d two dimensional random variables under the conditioning that they form a record under mild conditions we show that all random variables tend to concentrate as n rightarrow infty around limiting curves which are the solutions of an appropriate variational problem we also show that the same phenomenon occurs without the records conditioning for the longest increasing subsequence in the sample

discrete series characters and fourier inversion on semisimple real lie groups let g be a semisimple real lie group explicit formulas for discrete series characters on noncompact cartan subgroups are given these formulas are used to give a simple formula for the fourier transform of orbital integrals of regular semisimple orbits

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

integrals over grassmannians and random permutations testing the independence of two gaussian populations involves the distribution of the sample canonical correlation coefficients given that the actual correlation is zero the laplace transform as a function of x of this distribution is not only an integral over the grassmannian gr p f n of p dimensional planes in real complex or quaternion n space f n but is also related to a generalized hypergeometric function such integrals are solutions of painleve like equations in the complex case they are solutions to genuine painleve equations these integrals over gr p c n have remarkable expansions in x related to random words of length l formed with an alphabet of p letters 1 p the coefficients of these expansions are given by the probability that a word i contains a subsequence of letters p p 1 1 in that order and ii that the maximal length of the disjoint union of p 1 increasing subsequences of the word is less than or equal tok where k refers to the power of x note that if each letter appears in the word then the maximal length of the disjoint union of p increasing subsequences of the word is automatically l and is thus trivial c 2003 elsevier inc all rights reserved

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

gaussian fluctuations of jack deformed random young diagrams we introduce a large class of random young diagrams which can be regarded as a natural one parameter deformation of some classical young diagram ensembles a deformation which is related to jack polynomials and jack characters we show that each such a random young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically gaussian

raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non homogeneous lattices we complete the construction of raising and lowering operators given in a previous work for the orthogonal polynomials of hypergeometric type on a non homogeneous lattice we extend these operators to the generalized orthogonal polynomials namely those difference orthogonal polynomials that satisfy a similar difference equation of hypergeometric type

affine shuffles shuffles with cuts the whitehouse module and patience sorting abstract type a affine shuffles are compared with riffle shuffles followed by a cut although these probability measures on the symmetric group s n are different they both satisfy a convolution property strong evidence is given that when the underlying parameter q satisfies gcd n q 1 1 the induced measures on conjugacy classes of the symmetric group coincide this gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents using representation theoretic work on the whitehouse module a formula is obtained for the cycle structure of a riffle shuffle followed by a cut it is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness generating functions for the first pile size in patience sorting from decks with repeated values are derived this relates to random matrices

a note on the diffusion of directed polymers in a random environment a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford 3 and if the disorder is small enough this simplifies and extends an approach by j imbrie and t spencer

limiting shape for directed percolation models we consider directed first passage and last passage percolation on the nonnegative lattice z d d 2 with i i d weights at the vertices under certain moment conditions on the common distribution of the weights the limits g x lim n n 1 t nx exist and are constant a s for x r d where t z is the passage time from the origin to the vertex z z d we show that this shape function g is continuous on r d in particular at the boundaries in two dimensions we give more precise asymptotics for the behavior of g near the boundaries these asymptotics depend on the common weight distribution only through its mean and variance in addition we discuss growth models which are naturally associated to the percolation processes giving a shape theorem and illustrating various possible types of behavior with output from simulations

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

random walk generated by random permutations of 1 2 3 n 1 we study properties of a non markovian random walk x n l l 0 1 2 n evolving in discrete time l on a one dimensional lattice of integers whose moves to the right or to the left are prescribed by the rise and descent sequences characterizing random permutations of n 1 1 2 3 n 1 we determine exactly the probability of finding the end point xn x n n of the trajectory of such a permutation generated random walk pgrw at site x and show that in the limit n it converges to a normal distribution with a smaller compared to the conventional polya random walk diffusion coefficient we formulate as well an auxiliary stochastic process whose distribution is identical to the distribution of the intermediate points x n l l n which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of turns of the pgrw trajectories

scaling limit for the space time covariance of the stationary totally asymmetric simple exclusion process the totally asymmetric simple exclusion process tasep on the one dimensional lattice with the bernoulli measure as initial conditions 0 1 is stationary in space and time let nt j be the number of particles which have crossed the bond from j to j 1 during the time span 0 t for open image in new window we prove that the fluctuations of nt j for large t are of order t1 3 and we determine the limiting distribution function open image in new window which is a generalization of the gue tracy widom distribution the family open image in new window of distribution functions have been obtained before by baik and rains in the context of the png model with boundary sources which requires the asymptotics of a riemann hilbert problem in our work we arrive at open image in new window through the asymptotics of a fredholm determinant open image in new window is simply related to the scaling function for the space time covariance of the stationary tasep equivalently to the asymptotic transition probability of a single second class particle

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

asymptotic results on infinite tandem queueing networks we consider an infinite tandem queueing network consisting of gi 1 stations with i i d service times we investigate the asymptotic behavior of t n k the inter arrival times between customers n and n 1 at station k and that of w n k the waiting time of customer n at station k we establish a duality property by which w n k and the idle times y n k play symmetrical roles this duality structure interesting by itself is also instrumental in proving some of the ergodic results we consider two versions of the model the quadrant and the half plane in the quadrant version the sequences of boundary conditions w 0 k k and t n 0 n are given in the half plane version the sequence t n 0 n is given under appropriate assumptions on the boundary conditions and on the services we obtain ergodic results for both versions of the model for the quadrant version we prove the existence of temporally ergodic evolutions and of spatially ergodic ones furthermore the process t n k n converges weakly with k to a limiting distribution which is invariant for the queueing operator in the more difficult half plane problem the aim is to obtain evolutions which are both temporally and spatially ergodic we prove that 1 n k 1 nw 0 k converges almost surely and in l1 to a finite constant this constitutes a first step in trying to prove that t n k n converges weakly with k to an invariant limiting distribution

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

noninvariant integrals on semisimple groups of r rank one integrales invariantes et non invariantes definition et proprietes integrales d eisenstein et paquets d onde quelques estimations elementaires application aux asymptotes le role du terme constant utilisation des equations differentielles operateurs d entrelacement et c fonctions definition et proprietes de la transformee de fourier interpretation des calculs elimination de la k finitude isolation de centralite passage a la limite

the double torus as a 2d cosmos groups geometry and closed geodesics the double torus provides a relativistic model for a closed 2d cosmos with topology of genus 2 and constant negative curvature its unfolding into an octagon extends to an octagonal tessellation of its universal covering the hyperbolic space h2 the tessellation is analysed with tools from hyperbolic crystallography actions on h2 of groups subgroups are identified for su 1 1 for a hyperbolic coxeter group acting also on su 1 1 and for the homotopy group 2 whose extension is normal in the coxeter group closed geodesics arise from links on h2 between octagon centres the direction and length of the shortest closed geodesics is computed dedicated to marcos moshinsky on the occasion of his 80th birthday

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

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

numerical schemes and rates of convergence for the hamilton jacobi equation continuum limit of nondominated sorting non dominated sorting arranges a set of points in n dimensional euclidean space into layers by repeatedly removing the coordinatewise minimal elements it was recently shown that nondominated sorting of random points has a hamilton jacobi equation continuum limit the obvious numerical scheme for this pde has a slow convergence rate of o h frac 1 n o h1n in this paper we introduce two new numerical schemes that have formal rates of o h and we prove the usual o sqrt h o h theoretical rates we also present the results of numerical simulations illustrating the difference between the formal and theoretical rates

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

geodesics and spanning trees for euclidean first passage percolation the metric d q q on the set q of particle locations of a homogeneous poisson process on r d defined as the infimum of i q i q i 1 1 over sequences in q starting with q and ending with q where denotes euclidean distance has nontrivial geodesics when a 1 the cases 1 are the euclidean first passage percolation fpp models introduced earlier by the authors while the geodesics in the case are exactly the paths from the euclidean minimal spanning trees forests of aldous and steele we compare and contrast results and conjectures for these two situations new results for 1 and any d include inequalities on the fluctuation exponents for the metric x 1 2 and for the geodesies 3 4 in strong enough versions to yield conclusions not yet obtained for lattice fpp almost surely every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic from every q for d 2 and 2 further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces

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

a pde based approach to nondominated sorting nondominated sorting is a fundamental combinatorial problem in multiobjective optimization and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science in a previous work siam j math anal 46 2014 pp 603 638 we showed that nondominated sorting has a continuum limit that corresponds to solving a hamilton jacobi equation in this work we present and analyze a fast numerical scheme for this hamilton jacobi equation and show how it can be used to design a fast algorithm for approximate nondominated sorting

current fluctuations for tasep a proof of the prahofer spohn conjecture we consider the family of two sided bernoulli initial conditions for tasep which as the left and right densities p are varied give rise to shock waves and rarefaction fans the two phenomena which are typical to tasep we provide a proof of conjecture 7 1 of progr probab 51 2002 185 204 which characterizes the order of and scaling functions for the fluctuations of the height function of two sided tasep in terms of the two densities and the speed y around which the height is observed in proving this theorem for tasep we also prove a fluctuation theorem for a class of comer growth processes with external sources or equivalently for the last passage time in a directed last passage percolation model with two sided boundary conditions and 1 p we provide a complete characterization of the order of and the scaling functions for the fluctuations of this model s last passage time l n m as a function of three parameters the two boundary source rates p and 1 p and the scaling ratio y 2 m n the proof of this theorem draws on the results of comm math phys 265 2006 1 44 and extensively on the work of ann probab 33 2005 1643 1697 on finite rank perturbations of wishart ensembles in random matrix theory

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

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

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

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

on the long time behavior of the doubly infinite toda lattice under initial data decaying at infinity we provide rigorous analysis of the long time behavior of the doubly infinite toda lattice under initial data that decay at infinity in the absence of solitions we solve approximately and for large times the riemann hilbert matrix factorization problem equivalent to the related inverse scattering problem with the help of the beals coifman formula by reducing it to a simpler one through a series of contour deformations in the spirit of the deift zhou method

lengths of monotone subsequences in a mallows permutation we study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the mallows measure under this measure the probability of a permutation uppi in s n is proportional to q mathrm inv uppi where q is a real parameter and mathrm inv uppi is the number of inversions in uppi the case q 1 corresponds to uniformly random permutations the mallows measure was introduced by mallows in connection with ranking problems in statistics we determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences as well as large deviation bounds for them we also provide a simple bound on the variance of these lengths and prove a law of large numbers for the length of the longest increasing subsequence assuming without loss of generality that q 1 our results apply when q is a function of n satisfying n 1 q rightarrow infty the case that n 1 q o 1 was considered previously by mueller and starr in our parameter range the typical length of the longest increasing subsequence is of order n sqrt 1 q whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q we show also that in the graphical representation of a mallows distributed permutation most points are found in a symmetric strip around the diagonal whose width is of order 1 1 q this suggests a connection between the longest increasing subsequence in the mallows model and the model of last passage percolation in a strip

a direct verification argument for the hamilton jacobi equation continuum limit of nondominated sorting abstract nondominated sorting is a combinatorial algorithm that sorts points in euclidean space into layers according to a partial order it was recently shown that nondominated sorting of random points has a hamilton jacobi equation continuum limit the original proof given in calder et al 2014 relies on a continuum variational problem in this paper we give a new proof using a direct verification argument that completely avoids the variational interpretation we believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle

statistical self similarity of one dimensional growth processes abstract for one dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times we argue that for self similar growth from a single seed the universal distribution is the tracy widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other only numerically determined distribution in particular for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation from which the height distribution is identified as the tracy widom distribution

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

integral formulas for the asymmetric simple exclusion process in this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice mathbb z with nearest neighbor hopping rates p to the right and q 1 p to the left for the most part we consider an n particle system but for certain of these formulas we can take the n to infty limit first we obtain for the n particle system a formula for the probability of a configuration at time t given the initial configuration for this we use bethe ansatz ideas to solve the master equation extending a result of schutz for the case n 2 the main results of the paper derived from this are integral formulas for the probability for given initial configuration that the mth left most particle is at x at time t in one of these formulas we can take the n to infty limit and it gives the probability for an infinite system where the initial configuration is bounded on one side for the special case of the totally asymmetric simple exclusion process tasep our formulas reduce to the known ones

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

painleve ii asymptotics near the leading edge of the oscillatory zone for the korteweg de vries equation in the small dispersion limit in the small dispersion limit solutions to the korteweg de vries equation develop an interval of fast oscillations after a certain time we obtain a universal asymptotic expansion for the korteweg de vries solution near the leading edge of the oscillatory zone up to second order corrections this expansion involves the hastings mcleod solution of the painleve ii equation we prove our results using the riemann hilbert approach c 2009 wiley periodicals inc

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

a new parametrization and all integral realizations of the lorentz group a simple linear transformation of the biquaternionic parameters and of the group sl 2 c which is two to one homomorphic to the restricted lorentz group l is used to express each element of sl 2 c and l in terms of the first column of an element of l and the quaternionic parameters this parametrization is shown to bring the determination of the integral realizations of the restricted lorentz group under the purview of classical diophantine analysis involving the expression of a given pair of integers respectively as a sum of three squares and a sum of four squares these solutions are further constrained by the condition that a set of four integers which is linear in the solution of the three squares problem and in the solution of the four squares problem have a common factor these are all classical problems addressed and solved by euclid euler fermat gauss jacobi and others the corresponding realizations of sl 2 c fall into three distinct classes those in which the elements of

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

the random average process and random walk in a space time random environment in one dimension we study space time fluctuations around a characteristic line for a one dimensional interacting system known as the random average process the state of this system is a real valued function on the integers new values of the function are created by averaging previous values with random weights the fluctuations analyzed occur on the scale n1 4 where n is the ratio of macroscopic and microscopic scales in the system the limits of the fluctuations are described by a family of gaussian processes in cases of known product form invariant distributions this limit is a two parameter process whose time marginals are fractional brownian motions with hurst parameter 1 4 along the way we study the limits of quenched mean processes for a random walk in a space time random environment these limits also happen at scale n1 4 and are described by certain gaussian processes that we identify in particular when we look at a backward quenched mean process the limit process is the solution of a stochastic heat equation

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