asymptotics in asep with step initial condition in previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition particles beginning at the positive integers there it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a fredholm determinant here we use this formula to obtain three asymptotic results for the positions of these particles in one an apparently new distribution function arises and in another the distribution function f2 arises the latter extends a result of johansson on tasep to asep and hence proves kpz universality for asep with step initial condition

on the joint distribution of the maximum and its position of the airy2 process minus a parabola the maximal point of the airy2 process minus a parabola is believed to describe the scaling limit of the end point of the directed polymer in a random medium this was proved to be true for a few specific cases recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained one by moreno flores quastel and remenik and the other by schehr the first formula is given in terms of the airy function and an associated operator and the second formula is expressed in terms of the lax pair equations of the painleve ii equation we give a direct proof that these two formulas are the same

universality class of replica symmetry breaking scaling behavior and the low temperature fixed point order function of the sherrington kirkpatrick model a scaling theory of replica symmetry breaking rsb in the sherrington kirkpatrick sk model is presented in the framework of critical phenomena for the scaling regime of large rsb orders ensuremath kappa small temperatures t and small homogeneous magnetic fields h we employ the pseudodynamical picture r oppermann m j schmidt and d sherrington phys rev lett 98 127201 2007 where two critical points mathcal c mathcal p 1 and mathcal c mathcal p 2 are associated with the order function s pseudodynamical limits mathrm lim a ensuremath rightarrow ensuremath infty phantom rule 0 2em 0ex q a 1 and mathrm lim a ensuremath rightarrow 0 phantom rule 0 2em 0ex q a 0 at t 0 h 0 1 ensuremath kappa 0 mathcal c mathcal p 1 and mathcal c mathcal p 2 dominated contributions to the free energy functional f q a require an unconventional scaling hypothesis we determine the scaling contributions in accordance with detailed numerical self consistent solutions for up to 200 orders of rsb power laws scaling functions and crossover lines are obtained mathcal c mathcal p 1 dominated behavior is found for the nonequilibrium susceptibility which decays like ensuremath chi 1 ensuremath kappa ensuremath 5 3 f 1 t ensuremath kappa ensuremath 5 3 for the entropy which obeys s t 0 ensuremath sim ensuremath chi 1 2 and for the subclass of diverging parameters a i ensuremath kappa 5 3 f a i t ensuremath kappa ensuremath 5 3 describing parisi box sizes m i t ensuremath equiv a i t t with f 1 ensuremath zeta ensuremath sim ensuremath zeta and f a i ensuremath zeta ensuremath sim 1 ensuremath zeta for ensuremath zeta ensuremath rightarrow ensuremath infty while f 0 is finite mathcal c mathcal p 2 dominated behavior controlled by the magnetic field h while temperature is irrelevant is retrieved in the plateau height or width of the order function q a according to q pl h ensuremath kappa ensuremath 1 f pl h 2 3 ensuremath kappa ensuremath 1 with f pl phantom ensuremath mid ensuremath zeta ensuremath mid ensuremath zeta ensuremath rightarrow ensuremath infty ensuremath sim ensuremath zeta and f pl 0 finite divergent characteristic rsb orders ensuremath kappa mathcal c mathcal p 1 t ensuremath sim t ensuremath 3 5 and ensuremath kappa mathcal c mathcal p 2 h ensuremath sim h ensuremath 2 3 respectively describe the crossover from mean field sk to rsb critical behavior with rational valued exponents extracted with high precision from our rsb data the order function q a is obtained as a fixed point function q a of rsb flow in agreement with integrated fixed point energy and susceptibility distributions

geodesics toward corners in first passage percolation for stationary first passage percolation in two dimensions the existence and uniqueness of semi infinite geodesics directed in particular directions or sectors has been considered by damron and hanson 4 ahlberg and hoffman 1 and others however the main results do not cover geodesics in the direction of corners of the limit shape b where two facets meet we construct an example with the following properties i the limiting shape is an octagon ii semi infinite geodesics exist only in the four axis directions and iii in each axis direction there are multiple such geodesics consequently the set of points of b which are in the direction of some geodesic does not have all of b as its convex hull

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

kardar parisi zhang universality class and the anchored toom interface we revisit the anchored toom interface and use kardar parisi zhang scaling theory to argue that the interface fluctuations are governed by the airy1 process with the role of space and time interchanged the predictions which contain no free parameter are numerically well confirmed for space time statistics in the stationary state in particular the spatial fluctuations of the interface computed numerically agree well with those given by the goe edge distribution of tracy and widom

on the zeros of the q analogue exponential function an asymptotic formula for the zeros zn of the entire function eq x for q 1 is obtained as q increases above the first collision point at q1 approximately 0 14 these zeros collide in pairs and then move off into the complex z plane they move off as and remain a complex conjugate pair the zeros of the ordinary higher derivatives and of the ordinary indefinite integrals of eq x vary with q in a similar manner properties of eq z for z complex and for arbitrary q are deduced for 0 or q 1 eq z is an entire function of order 0 by the hadamard weierstrass factorization theorem infinite product representations are obtained for eq z and for the reciprocal function eq 1 z if q not 1 the zeros satisfy the sum rule sigma n 1infinity 1 zn 1

the arctic circle boundary and the airy process we prove that the appropriately rescaled boundary of the north polar region in the aztec diamond converges to the airy process the proof uses certain determinantal point processes given by the extended krawtchouk kernel we also prove a version of propp s conjecture concerning the structure of the tiling at the center of the aztec diamond 1 introduction and results domino tilings of the aztec diamond were introduced in 8 9 asymptotic properties of random domino tilings of the aztec diamond have been studied in 5 12 15 in particular in 12 the existence of the so called arctic circle was proved the arctic circle is the asymptotic boundary of the disordered so called temperate region of the tiling outside this boundary the tiling forms a completely regular brick wall pattern the methods in 12 combined with the results in 13 show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the tracy widom distribution of random matrix theory in this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the airy process introduced in 23 the paper is a continuation of the approach used in 14 and 15 where certain point processes with determinantal correlation functions 24 and the krawtchouk ensemble were used we will use the general techniques developed in 16 and investigate an extended point process which also has determinantal correlation functions given by a kernel which we call the extended krawtchouk kernel the aztec diamond an of order n is the union of all lattice squares m m 1 l l 1 m l 2 z that lie inside the region x1 y1 x1 y1

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

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

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

a hybrid method combining the multitemporal resolution time domain method of moments with the time domain geometrical theory of diffraction for thin wire antenna problems in this paper a hybrid method combining a multitemporal resolution mtr enhanced time domain method of moments td mom with the td geometrical theory of diffraction gtd is presented which allows to efficiently calculate the transient fields radiated by antennas in presence of large objects the mtr scheme tailors the time step size and thus implicitly the duration of the temporal basis function on the basis of the distance between source and test element additionally the voltage induced by source elements far away from the test element is interpolated in time the hybrid method is applied to calculate the radiation properties of thin wire antennas in presence of perfectly conducting flat scatterers to demonstrate its basic features and advantages

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

the airy1 process is not the limit of the largest eigenvalue in goe matrix diffusion using a systematic approach to evaluate fredholm determinants numerically we provide convincing evidence that the airy1 process arising as a limit law in stochastic surface growth is not the limit law for the evolution of the largest eigenvalue in goe matrix diffusion

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

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

busemann functions and infinite geodesics in two dimensional first passage percolation we study first passage percolation on mathbb z 2 where the edge weights are given by a translation ergodic distribution addressing questions related to existence and coalescence of infinite geodesics some of these were studied in the late 1990s by c newman and collaborators under strong assumptions on the limiting shape and weight distribution in this paper we develop a framework for working with distributional limits of busemann functions and use it to prove forms of newman s results under minimal assumptions for instance we show a form of coalescence of long finite geodesics in any deterministic direction we also introduce a purely directional condition which replaces newman s global curvature condition and whose assumption we show implies the existence of directional geodesics without this condition we prove existence of infinite geodesics which are directed in sectors last we analyze distributional limits of geodesic graphs proving almost sure coalescence and nonexistence of infinite backward paths this result relates to the conjecture of nonexistence of bigeodesics

solution of radiation and scattering problems in complex environments using a hybrid finite integration technique uniform theory of diffraction approach the finite integration technique fit is combined with the uniform geometrical theory of diffraction utd for the solution of radiation and scattering problems in complex environments the presented hybrid formulation is capable of handling large objects allowing in the same time a precise modeling of important geometrical details and material inhomogeneities the part of the structure which contains the details of the geometrical model and the material inhomogeneities is discretized and solved using the fit discretization scheme whereas the influence of the large scatterers to the total solution is resolved by means of utd in contrast with other finite difference based hybridizations in the presented formulation the case of the strong coupling between the two subproblems without simplifications is considered this is accomplished by introducing an appropriate boundary operator derived by the equivalence principle the resulting equation system possesses a complex nearly dense system matrix which is difficult to handle even using iterative solvers to overcome this difficulty the system of equations is solved using a two step procedure i e the fit equation and the boundary condition are treated separately

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

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

current distribution and random matrix ensembles for an integrable asymmetric fragmentation process we calculate the time evolution of a discrete time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size dependent rates in the continuous time limit the process turns into the totally asymmetric simple exclusion process only pieces of size 1 break off a given cluster we express the exact solution of the master equation for the process in terms of a determinant which can be derived using the bethe ansatz from this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the gaussian unitary ensemble of random matrices this result confirms universality of the scaling form of the current distribution in the kpz universality class and suggests that there is a link between integrable particle systems and random matrix ensembles

crossover distributions at the edge of the rarefaction fan we consider the weakly asymmetric limit of simple exclusion process with drift to the left starting from step bernoulli initial data with so that macroscopically one has a rarefaction fan we study the fluctuations of the process observed along slopes in the fan which are given by the hopf cole solution of the kardar parisi zhang kpz equation with appropriate initial data for slopes strictly inside the fan the initial data is a dirac delta function and the one point distribution functions have been computed in comm pure appl math 64 2011 466 537 and nuclear phys b 834 2010 523 542 at the edge of the rarefaction fan the initial data is one sided brownian we obtain a new family of crossover distributions giving the exact one point distributions of this process which converge as t to those of the airy a2 bm process as an application we prove moment and large deviation estimates for the equilibrium hopf cole solution of kpz these bounds rely on the apparently new observation that the fkg inequality holds for the stochastic heat equation finally via a feynman kac path integral the kpz equation also governs the free energy of the continuum directed polymer and thus our formula may also be interpreted in those terms

asymptotic shapes for stationary first passage percolation this paper deals with first passage percolation where the usual i i d condition is weakened to stationarity and ergodicity the well known asymptotic shape result is known to extend to this case it is easy to give necessary conditions for a compact set b subset mathbb r d to arise as the asymptotic shape for some stationary measure on the passage times our main result says that these conditions are also sufficient

a pfaffian representation for flat asep we obtain a fredholm pfaffian formula for an appropriate generating function of the height function of the asymmetric simple exclusion process starting from flat periodic initial data formal asymptotics lead to the goe tracy widom distribution 2016 wiley periodicals inc

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

a hybrid febi mlfmm utd method for numerical solutions of electromagnetic problems including arbitrarily shaped and electrically large objects numerical solutions of electromagnetic scattering and radiation problems including arbitrarily shaped objects are obtained by solving integral equations with the method of moments mom fast and efficient solution of the integral equation with low computation and memory complexity is provided by the multilevel fast multipole method mlfmm the presence of electrically large conducting objects leads to hybrid mom techniques with high frequency methods for ray based high frequency methods no discretization of the electrically large objects is needed resulting into a more efficient numerical treatment of the problem however in order to retain low computation and memory complexity the high frequency fields must be taken into account in the matrix vector product computations in the various levels of the mlfmm in this contribution a ray based hybridization of the mlfmm with the uniform geometrical theory of diffraction utd is proposed within a hybrid finite element boundary integral febi technique using the combined field integral equation cfie resulting into a hybrid febi mlfmm utd method the hybridization is performed at the translation procedure on the various levels of the mlfmm using a far field approximation of the appropriate translation operator to obtain the high frequency incident fields at the critical points of the utd the formulation of this new hybrid technique is presented and numerical results are shown

endpoint distribution of directed polymers in 1 1 dimensions we give an explicit formula for the joint density of the max and argmax of the airy2 process minus a parabola the argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1 1 dimensions

exact solution for the kardar parisi zhang equation with flat initial conditions we provide the first exact calculation of the height distribution at arbitrary time t of the continuum kardar parisi zhang kpz growth equation in one dimension with flat initial conditions we use the mapping onto a directed polymer with one end fixed one free and the bethe ansatz for the replicated attractive boson model we obtain the generating function of the moments of the directed polymer partition sum as a fredholm pfaffian our formula valid for all times exhibits convergence of the free energy i e kpz height distribution to the gaussian orthogonal ensemble tracy widom distribution at large time

from duality to determinants for q tasep and asep we prove duality relations for two interacting particle systems the q deformed totally asymmetric simple exclusion process q tasep and the asymmetric simple exclusion process asep expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents respectively duality implies that they solve systems of odes these systems are integrable and for particular step and half stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems solutions and hence also the moments we form laplace transform like generating functions of these moments and via residue calculus we compute two different types of fredholm determinant formulas for such generating functions for asep the first type of formula is new and readily lends itself to asymptotic analysis as necessary to reprove gue tracy widom distribution fluctuations for asep while the second type of formula is recognizable as closely related to tracy and widom s asep formula comm math phys 279 2008 815 844 j stat phys 132 2008 291 300 comm math phys 290 2009 129 154 j stat phys 140 2010 619 634 for q tasep both formulas coincide with those computed via borodin and corwin s macdonald processes probab theory related fields 2014 158 225 400 both q tasep and asep have limit transitions to the free energy of the continuum directed polymer the logarithm of the solution of the stochastic heat equation or the hopf cole solution to the kardar parisi zhang equation thus q tasep and asep are integrable discretizations of these continuum objects the systems of odes associated to their dualities are deformed discrete quantum delta bose gases and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics

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

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

stationary stochastic higher spin six vertex model and q whittaker measure in this paper we consider the higher spin six vertex model on the lattice mathbb z ge 2 times mathbb z ge 1 we first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space time limit for the stationary higher spin six vertex model in particular along the characteristic line we recover baik rains fluctuations with size of characteristic exponent 1 3 we also consider some of the main degenerations of the higher spin six vertex model and we adapt our analysis to the relevant cases of the q hahn particle process and of the exponential jump model

a fredholm determinant representation in asep in previous work tracy and widom in commun math phys 279 815 844 2008 the authors found integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice the dynamics are uniquely determined once the initial state is specified in this note we restrict our attention to the case of step initial condition with particles at the positive integers and consider the distribution function for the mth particle from the left in tracy and widom commun math phys 279 815 844 2008 an infinite series of multiple integrals was derived for the distribution in this note we show that the series can be summed to give a single integral whose integrand involves a fredholm determinant we use this determinant representation to derive non rigorously at this writing a scaling limit

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

on asep with step bernoulli initial condition this paper extends earlier work on asep to the case of step bernoulli initial condition the main results are a representation in terms of a fredholm determinant for the probability distribution of a fixed particle and asymptotic results which in particular establish kpz universality for this probability in one regime and as a corollary for the current fluctuations

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

mutually avoiding paths in random media and largest eigenvalues of random matrices recently it was shown that the probability distribution function pdf of the free energy of a single continuum directed polymer dp in a random potential equivalently to the height of a growing interface described by the kardar parisi zhang kpz equation converges at large scale to the tracy widom distribution the latter describes the fluctuations of the largest eigenvalue of a random matrix drawn from the gaussian unitary ensemble gue and the result holds for a dp with fixed end points i e for the kpz equation with droplet initial conditions a more general conjecture can be put forward relating the free energies of n 1 noncrossing continuum dp in a random potential to the sum of the nth largest eigenvalues of the gue here using replica methods we provide an important test of this conjecture by calculating exactly the right tails of both pdfs and showing that they coincide for arbitrary n

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

stationary correlations for the 1d kpz equation we study exact stationary properties of the one dimensional kardar parisi zhang kpz equation by using the replica approach the stationary state for the kpz equation is realized by setting the initial condition the two sided brownian motion bm with respect to the space variable developing techniques for dealing with this initial condition in the replica analysis we elucidate some exact nature of the height fluctuation for the kpz equation in particular we obtain an explicit representation of the probability distribution of the height in terms of the fredholm determinants furthermore from this expression we also get the exact expression of the space time two point correlation function

fully coupled hybrid fem utd method using nurbs for the analysis of radiation problems a novel hybrid finite element method fem uniform theory of diffraction utd method for the analysis of radiating structures in the presence of electrically large objects is presented the hybridization is done in a fully coupled way taking into account mutual interactions between the fem and utd regions the utd objects are modeled using non uniform rational b spline nurbs surfaces ray acceleration techniques and simple clustering strategies are used to speed up the computation time numerical results are presented showing the validity of the method

vicious random walkers in the limit of a large number of walkers the vicious random walker problem on a line is studied in the limit of a large number of walkers the multidimensional integral representing the probability that thep walkers will survive a timet denotedp is shown to be analogous to the partition function of a particular one component coulomb gas by assuming the existence of the thermodynamic limit for the coulomb gas one can deduce asymptotic formulas forp in the large p large t limit a straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration this event is termed a reunion consequently asymptotic formulas for the conditional probability of a reunion given that all walkers survive are derived also an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet given that allp walkers survive is calculated in the limitt p

first passage percolation the stationary case if the passage time of the edges of the d lattice are stationary ergodic and have finite moment of orderp d then a s the set of vertices that can be reached within timet has an asymptotic shape ast

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

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

hybrid sup 2 combining the three dimensional hybrid finite element boundary integral technique for planar multilayered media with the uniform geometrical theory of diffraction the fully three dimensional 3 d hybrid finite element fe boundary integral bi technique is extended by further hybridization with the uniform geometrical theory of diffraction utd resulting in a so called hybrid sup 2 fe bi utd approach the formulation is capable of modeling arbitrarily shaped strongly inhomogeneous objects together with electrically large obstacles of relatively simple shape within the common environment of a planar multilayered medium the arbitrarily shaped inhomogeneous objects are discretized by finite elements whereas the electrically large obstacles are described by the utd and both of these models are included into an integral equation derived from the equivalence principle for planar multilayered media thus full electromagnetic coupling is realized between all parts of the formulation the integral equation is implemented using mixed potentials with appropriate green s functions derived from sommerfeld integral representations for planar multilayered media the utd contributions are accounted for by corresponding modifications of the green s functions and the fe technique is coupled to the integral equation via introduction of equivalent surface current densities in the bounding surfaces of the discretized objects after presenting the formulation of this novel fully 3 d hybrid sup 2 technique the implemented computer code is validated against conventional hybrid fe bi computations and a wireless base station antenna is analyzed in several situations of complex real world microcell environments

fredholm determinants and pole free solutions to the noncommutative painleve ii equation we extend the formalism of integrable operators a la its izergin korepin slavnov to matrix valued convolution operators on a semi infinite interval and to matrix integral operators with a kernel of the form frac e 1 t lambda e 2 mu lambda mu thus proving that their resolvent operators can be expressed in terms of solutions of some specific riemann hilbert problems we also describe some applications mainly to a noncommutative version of painleve ii recently introduced by retakh and rubtsov and a related noncommutative equation of painleve type we construct a particular family of solutions of the noncommutative painleve ii that are pole free for real values of the variables and hence analogous to the hastings mcleod solution of commutative painleve ii such a solution plays the same role as its commutative counterpart relative to the tracy widom theorem but for the computation of the fredholm determinant of a matrix version of the airy kernel

a determinantal formula for the goe tracy widom distribution investigating the long time asymptotics of the totally asymmetric simple exclusion process sasamoto obtains rather indirectly a formula for the tracy widom distribution for the gaussian orthogonal ensemble we establish that his novel formula indeed agrees with more standard expressions

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

anomalous shock fluctuations in tasep and last passage percolation models we consider the totally asymmetric simple exclusion process with initial conditions and or jump rates such that shocks are generated if the initial condition is deterministic then the shock at time t will have a width of order t 1 3 we determine the law of particle positions in the large time limit around the shock in a few models in particular we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the airy 1 process the limiting distribution is a product of two distribution functions which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics we show that the result generalizes to generic last passage percolation models

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

finite element analysis of complex antennas and arrays various technical issues are addressed in the frequency and time domain finite element analysis of complex antennas and arrays that may consist of fine structures and composite materials the paper starts with the finite element formulation of a generic antenna problem formulation for the modeling of general electrically and magnetically dispersive lossy and anisotropic materials in the time domain finite element analysis is presented for the first time this is followed by discussion of three typical techniques for truncation of the finite element computational domain via the use of absorbing boundary conditions perfectly matched layers and boundary integral equations it then describes the modeling of antenna feeds using a current probe including an improved model a voltage gap and a waveguide port boundary condition after that numerical approaches to modeling infinite phased arrays large finite arrays and antenna platform interaction are presented including two novel domain decomposition algorithms in both the frequency and time domains numerical examples are given to demonstrate the capability and application of the finite element analysis of a variety of complex antennas and arrays

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