uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

on the finite size scaling of clusters in compact directed percolation the exact finite size scaling properties of clusters in compact directed percolation on a square lattice are derived the results are implicit in previous work on the enumeration of staircase polygons but their explicit form has not been presented as such before the analysis provides important insights into the nature of this type of percolation transition

scaling analysis for the adsorption transition in a watermelon network of n directed non intersecting walks the partition function for the problem of n directed non intersecting walks interacting via contact potentials with a wall parallel to the direction of the walks has previously been calculated as an n n determinant here we describe how to analyse the scaling behaviour of this problem using alternative representations of the solution in doing so we derive the asymptotics of the partition function of a watermelon network of n such walks for all temperatures and so calculate the associated network exponents in the three regimes desorbed adsorbed and at the adsorption transition furthermore we derive the full scaling function around the adsorption transition for all n at the adsorption transition we also derive a simple product form for the partition function these results have in part been derived using recurrence relations satisfied by the original determinantal solution

on the hulls of directed percolation clusters the properties of the hulls of directed percolation clusters are studied the scaling and finite size scaling of many quantities around the percolation threshold pc are derived and a novel monte carlo algorithm which is more than twice as fast as the standard algorithm at pc has been formulated to study these properties simulations have been conducted that enable an estimation of all the exponents involved in particular the central exponent x relating the average hull length of clusters to their mass has been estimated to be 0 773 4 at the percolation threshold this same exponent is estimated to be 0 905 5 for p p c thus this second value should hold for directed animals

plane partition vesicles we examine partitions and their natural three dimensional generalizations plane partitions as models of vesicles undergoing an inflation deflation transition the phase diagrams of these models include a critical point corresponding to an inflation deflation transition and exhibits multicritical scaling in the vicinity of a multicritical point located elsewhere on the critical curve we determine the locations of the multicritical points by analysing the generating functions using analytic and numerical means in addition we determine the numerical values of the multicritical scaling exponents associated with the multicritical scaling regimes in these models

variational identities and applications to differential systems j serrin 1 introduction in the work of ponos p and in recent work of pucci serrin ps variational identities are discussed which are useful for solving various ques tions about elliptic differential equations and particular elliptic systems here we obtain a number of further applications of these identities to systems it was noted in ps that these identities are closely related to noether s theorem for variational problems by working out the divergence term arising in noether s theorem and making some simplifications one obtains a general identity for variational problems if a variational integral has a symmetry group g with the following infinitesimal generator

polynomial inverse images and polynomial inequalities valid for polynomials pn of degree at most n in this paper we are primarily interested in what form these inequalities take on several intervals we shall see that the extension to general sets involves the equilibrium measure of these sets we shall give the precise form of the bernstein inequality for arbi trary compacts and an asymptotical ly best form of the markoff inequality for sets consisting of finitely many intervals actually in this case we shall prove different markoff inequalities one one associated with each one of the endpoints of the system of intervals the proofs will heavily use sets that are obtained as the inverse images of intervals under special polynomial mappings we shall see that the original bernstein and

some applications of the pohozaev identity we apply the pohozaev identity to show the nonexistence of nontrivial solutions to a semilinear equation of the form h e u f x u u where e is a real constant lying on the essential spectrum of h the self adjoint operators h under consideration are the schrodinger operator with coulomb type potentials the stark like hamiltonian and the semirelativistic hamiltonian

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

the height of watermelons with wall we derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall this generalizes a famous result of de bruijn et al 1972 graph theory and computing new york academic pp 15 22 on the average height of planted plane trees and results by fulmek 2007 electron j combin 14 r64 and katori et al 2008 j stat phys 131 1067 83 on the expected value and higher moments respectively of the height distribution of watermelons with two branches the asymptotics for the moments depend on the analytic behaviour of certain multidimensional dirichlet series in order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of jacobi s theta functions which generalizes the well known reciprocity law for jacobi s theta functions

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

the brownian excursion area a numerical analysis abstract using an explicit expression for the laplace transform of the brownian excursion area generating function a numerical analysis gives moments density and distribution function an asymptotic formula is given for small argument

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

nonequilibrium stationary states and equilibrium models with long range interactions it was recently suggested by blythe and evans that a properly defined steady state normalization factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities in analogy with the lee yang description of phase transition of equilibrium systems they studied the zeros in the complex plane of the normalization factor in order to find phase transitions in nonequilibrium steady states we show that like for equilibrium systems the densities associated with the rates are nondecreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the densities we illustrate this phenomenon for the asymmetric exclusion process we actually show that its normalization factor coincides with an equilibrium partition function of a walk model in which the densities have a simple physical interpretation

maximum distributions of bridges of noncolliding brownian paths one dimensional brownian motion starting from the origin at time t 0 conditioned to return to the origin at time t 1 and to stay positive during time interval 0 t 1 is called the bessel bridge with duration 1 we consider an n particle system of such bessel bridges conditioned never to collide with each other in 0 t 1 which is the continuum limit of the vicious walk model in watermelon configuration with a wall distributions of maximum values of paths attained in the time interval t epsilon 0 1 are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane for the outermost path the distribution function of maximum value is exactly determined for general n we show that the present n path system of noncolliding bessel bridges is realized as the positive eigenvalue process of the 2nx2n matrix valued brownian bridge in the symmetry class c using this fact computer simulations are performed and numerical results on the n dependence of the maximum value distributions of the inner paths are reported the present work demonstrates that the extreme value problems of noncolliding paths are related to random matrix theory the representation theory of symmetry and number theory

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

moments of vicious walkers and mobius graph expansions a system of brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval 0 t is studied the spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the gaussian unitary ensemble gue statistics to the gaussian orthogonal ensemble goe statistics as the time t goes on from 0 to t in the present paper we characterize this gue to goe transition by presenting the graphical expansion formula for the moments of positions of vicious walkers in the gue limit stackrel ensuremath rightarrow t 0 only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus following the time evolution of the vicious walkers however the graphs with twisted ribbons called m obius graphs increase their contribution to our expansion formula and we have to deal with the topology of nonorientable surfaces application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the m obius expansion using the stirling numbers of the first kind

extension of the arrowsmith essam formula to the domany kinzel model arrowsmith and essam gave an expansion formula for point to point connectedness functions of the mixed site bond percolation model on oriented lattices in which each term is characterized by a graph we extend this formula to general k point correlation functions which are point to set with k points connectivities in the context of percolation of the two neighbor discrete time markov process stochastic cellular automata with two parameters in one dimension called the domany kinzel model which includes the mixed site bond oriented percolation model on a square lattice as a special case our proof of the formula is elementary and based on induction with respect to time step which is different from the original graph theoretical one given by arrowsmith and essam we introduce a system of m interacting random walkers called m friendly walkers m fw with two parameters following the argument of cardy and colaiori it is shown that our formula is useful to derive a theorem that the correlation functions of the domany kinzel model are obtained as an m 0 limit of the generating functions of the m fw

vicious walkers and young tableaux i without walls we rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of young tableaux and integer partitions in this way we provide a proof of a result previously only conjectured for the total number of stars

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

finite length scaling of collapsing directed walks we study the finite length scaling of self interacting partially directed self avoiding walks utilising enumeration data up to a total length of 6000 steps this facilitates the evaluation of the numerical techniques available for calculating exponents at the point and in the collapsed phase of walk type models another consequence is the conjecture of a new scaling theory for the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of undirected problems including interacting saw we provide a phenomenological picture of the phase transition in terms of the condensation of droplets that allows us to understand the various length scales involved in the problem pacs numbers 05 50 q 05 70 fh 61 41 e email prel aleks brak tonyg mundoe maths mu oz au

scaling near the theta point for isolated polymers in solution recently questions have been raised as to the conclusions that can be drawn from currently proposed scaling theory for a single polymer in various types of solution in two and three dimensions here we summarize the crossover theory predicted for low dimensions and clarify the scaling arguments that relate thermal exponents for quantities on approaching the theta point from low temperatures to those associated with the asymptotics in polymer length at the theta point itself

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

the tricritical behavior of self interacting partially directed walks we present the thermodynamics of two variations of the interacting partially directed self avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values while the discrete model has been considered previously to varying degrees of success the continuous model we now define has not the examination of the continuous model leads to theexact derivation of several exponents for the discrete model some of these exponents can be calculated using a continued fraction representation for both models the crossover exponent is found to be 2 3 moreover we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function additionally the similarities noticed previously to other models but left unexplored are explained with the aid of necklacing arguments

uniform asymptotics of area weighted dyck paths using the generalized method of steepest descents for the case of two coalescing saddle points we derive an asymptotic expression for the bivariate generating function of dyck paths weighted according to their length and their area in the limit of the area generating variable tending towards 1 the result is valid uniformly for a range of the length generating variable including the tricritical point of the model

two bessel bridges conditioned never to collide double dirichlet series and jacobi theta function it is known that the moments of the maximum value of a one dimensional conditional brownian motion the three dimensional bessel bridge with duration 1 started from the origin are expressed using the riemann zeta function we consider a system of two bessel bridges in which noncolliding condition is imposed we show that the moments of the maximum value is then expressed using the double dirichlet series or using the integrals of products of the jacobi theta functions and its derivatives since the present system will be provided as a diffusion scaling limit of a version of vicious walker model the ensemble of 2 watermelons with a wall the dominant terms in long time asymptotics of moments of height of 2 watermelons are completely determined for the height of 2 watermelons with a wall the average value was recently studied by fulmek by a method of enumerative combinatorics

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

on toda lattices and orthogonal polynomials first we derive a simple connection between toda and langmuir lattices and give a characterization of toda lattices with the help of stieltjes functions then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic toda lattices the particles of the toda lattice are not even restricted as usual to move on the real line they may also move in the complex plane with the help of this result for special cases explicit solutions are obtained in terms of elliptic functions

directed compact percolation near a wall ii cluster length and size for paper i see ibid vol 27 p 3743 50 1994 the mean cluster size and length for the unbiassed growth of compact clusters near a dry wall are considered in the case of the cluster size below pc an exact expression is obtained for a seed of arbitrary width and distance from the surface it is found that the critical exponent gamma 1 for any finite distance from the surface crossover to the bulk value gamma 2 as the distance from the surface tends to infinity is observed this extends an existing result for the exponent beta of the percolation probability which changes from a value of 2 in the presence of a surface to 1 in the bulk limit the value delta 3 of the scaling size exponent is unchanged by the introduction of the surface the cluster size above pc and the mean cluster length are investigated using differential approximants from which we conjecture that these functions satisfy second order differential equations accepting this conjecture gives a mean size exponent the same as below pc and a logarithmic divergence of the mean length from both sides of the critical point the latter result together with scaling theory predicts that the exponent v sub has the value 2 the same as for the bulk problem

two step asymptotics of scaled dunkl processes dunkl processes are generalizations of brownian motion obtained by using the differential difference operators known as dunkl operators as a replacement of spatial partial derivatives in the heat equation special cases of these processes include dyson s brownian motion model and the wishart laguerre eigenvalue processes which are well known in random matrix theory it is known that the dynamics of dunkl processes is obtained by transforming the heat kernel using dunkl s intertwining operator it is also known that under an appropriate scaling their distribution function converges to a steady state distribution which depends only on the coupling parameter as the process time t tends to infinity we study scaled dunkl processes starting from an arbitrary initial distribution and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations in the first one is fixed and t tends to infinity approach to the steady s

exact solution of n directed non intersecting walks interacting with one or two boundaries the partition function for the problem of an arbitrary number of directed non intersecting walks interacting with one or two walls parallel to the direction of the walks is calculated exactly utilizing a theorem recently proved concerning the bethe ansatz for the eigenvectors of the transfer matrix of the five vertex model this theorem shows that the completeness of the bethe ansatz eigenvectors for the n walk transfer matrix can be deduced from the completeness of the one walk eigenvectors

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

cluster structure of collapsing polymers in order to better understand the geometry of the polymer collapse transition we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self avoiding walk we argue for this new correlated percolation problem that in two dimensions and possibly also in three dimensions a percolation transition takes place at a temperature lower than the collapse transition hence this novel transition should be governed by exponents unrelated to the point exponents this also implies that there is a temperature range in which the polymer has collapsed but has no long range cluster structure we use monte carlo to study the distribution of clusters on the simple cubic and manhattan lattices on the manhattan lattice where the data are most convincing we find that the percolation transition occurs at p 1 461 3 while the collapse transition is known to occur exactly at 1 414 we propose a finite size scaling form for the cluster distribution and estimate several of the critical exponents regardless of the value of p this percolation problem sheds new light on polymer collapse

return polynomials for non intersecting paths above a surface on the directed square lattice we enumerate sets of n non intersecting t step paths on the directed square lattice which are excluded from the region below the surface y 0t o which they are initially attached in particular we obtain a product formula for the number of star configurations in which the paths have arbitrary fixed endpoints we also consider the return polynomial r w y m0 r w y m m

scaling in the collapsed polymer phase exact results the author derives an exact scaling form for the low temperature partition function in a model of polymer collapse this confirms series work and so gives the exponents sigma 1 2 gamma 1 4 and chi 3 4 exactly the model considered is a variant of the self interacting partially directed self avoiding walk in two dimensions

shared information in stationary states of stochastic processes we present four estimators of the shared information or interdepency in ground states given that the coefficients appearing in the wave function are all real non negative numbers and therefore can be interpreted as probabilities of configurations such ground states of hermitian and non hermitian hamiltonians can be given for example by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models we consider in detail the last case the system being a classical not a quantum one using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive conformal invariant and nonconformal invariant massless phases we show that like in the case of the quantum problem the estimators verify the area law with logarithmic corrections when phase transitions take place

spectral analysis of n body stark hamiltonians we prove that the spectrum for a large class ofn body stark hamiltonians is purely absolutely continuous we need slow decay at infinity and local singularities of at most coulomb type in particular our results include the usual models for atoms and molecules

vicious walkers flows and directed percolation abstract it is shown that the problem of m vicious random walkers is equivalent to the enumeration of integer natural flows on a directed square lattice with maximum flow m an explicit formula for the number of such flows is given as a polynomial in m and is shown to have the same asymptotic form as fisher s determinantal result the expected number of such flows on directed bond or site percolation clusters is also a polynomial in m which reduces to the pair connectedness when m 0 consequently directed percolation may be seen as a problem of interacting vicious random walkers exact results for m 1 and 2 are given

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

rectangular vesicles in three dimensions the phase diagram of a model of rectangular vesicles in three dimensions is examined the scaling of the generating function is determined in the area perimeter volume area and volume perimeter ensembles the results are interpreted within the framework of tricritical scaling and the crossover exponents associated with the transitions are determined we identify three phases in the phase diagram a needle phase a disc phase and a cubical phase these phases are separated by three curves of transitions that meet in a multicritical point