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

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

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

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

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

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

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

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

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

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

spatial correlations of the 1d kpz surface on a flat substrate we study the spatial correlations of the one dimensional kpz surface for the flat initial condition it is shown that the multi point joint distribution for the height is given by a fredholm determinant with its kernel in the scaling limit explicitly obtained this may also describe the dynamics of the largest eigenvalue in the goe dyson s brownian motion model our analysis is based on a reformulation of the determinantal green s function for the totally asep in terms of a vicious walk problem

the growth of small droplets from a gas mixture results and estimates for very small radii abstract to test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture we develop a solution method especially suitable for radii smaller or not much larger than a mean free path the method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere for the simplest kinetic equation a variant of the linear bgk equation and for radii below two mean free paths this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent for radii around two mean free paths the new results agree with those of a variant of the moment method for lower radii discrepancies of up to a few percent are found our new method does not work for the klein kramers equation a kinetic equation for brownian particles for the latter equation we obtain an estimate for the current at very low radii from the free space solution by uhlenbeck and ornstein its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the bgk case

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

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

the kinetic boundary layer for the linearized boltzmann equation around an absorbing sphere we construct approximate solutions of the linearized boltzmann equation for a gas outside of a completely absorbing sphere a simple model for a liquid droplet growing in a supersaturated vapor the solutions are linear combinations of two chapman enskog type solutions which carry heat and particle currents and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path to construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere we expand the solution in burnett functions and truncate the resulting system of equations for the expansion coefficients for one particular truncation prescription which generalizes grad s 13 moment scheme good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere the results for small radii extrapolate smoothly toward the known limit of zero radius we present results for the reaction rate the particle current arriving at the sphere divided by the density at infinity and for the density and temperature profiles in the boundary layer the explicit calculations are carried out for maxwell molecules but the method appears to be suitable for more general intermolecular potentials

framelet based blind motion deblurring from a single image how to recover a clear image from a single motion blurred image has long been a challenging open problem in digital imaging in this paper we focus on how to recover a motion blurred image due to camera shake a regularization based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the motion blur kernel under tight wavelet frame systems furthermore an adapted version of the split bregman method is proposed to efficiently solve the resulting minimization problem the experiments on both synthesized images and real images show that our algorithm can effectively remove complex motion blurring from natural images without requiring any prior information of the motion blur kernel

directed polymers and the quantum toda lattice we characterize the law of the partition function of a brownian directed polymer model in terms of a diffusion process associated with the quantum toda lattice the proof is via a multidimensional generalization of a theorem of matsumoto and yor concerning exponential functionals of brownian motion it is based on a mapping which can be regarded as a geometric variant of the rsk correspondence

point to line polymers and orthogonal whittaker functions we study a one dimensional directed polymer model in an inverse gamma random environment known as the log gamma polymer in three different geometries point to line point to half line and when the polymer is restricted to a half space with end point lying free on the corresponding half line via the use of a n kirillov s geometric robinson schensted knuth correspondence we compute the laplace transform of the partition functions in the above geometries in terms of orthogonal whittaker functions thus obtaining new connections between the ubiquitous class of whittaker functions and exactly solvable probabilistic models in the case of the first two geometries we also provide multiple contour integral formulae for the corresponding laplace transforms passing to the zero temperature limit we obtain new formulae for the corresponding last passage percolation problems with exponential weights

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

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

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

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

scaling for a one dimensional directed polymer with boundary conditions we study a 1 1 dimensional directed polymer in a random environment on the integer lattice with log gamma distributed weights among directed polymers this model is special in the same way as the last passage percolation model with exponential or geometric weights is special among growth models namely both permit explicit calculations with appropriate boundary conditions the polymer with log gamma weights satisfies an analogue of burke s theorem for queues building on this we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path in the case where the boundary conditions are present and both endpoints of the polymer path are fixed for the polymer without boundary conditions and with either fixed or free endpoint we get the expected upper bounds on the exponents