adsorption of pulled directed paths in a slanted boundary in this paper the generating function free energy and phase diagram of a class of models of directed paths related to the models of paths adsorbing in the boundary y nx n 2 for and pulled by a vertical force in its endpoint are determined these models are related to the special classes of models of random walks in the quarter plane and which interact with the x axis generating functions are determined by solving a functional recurrence using the kernel method and are algebraic the generating function is determined for general values of n and the solutions for n 1 and n 2 are examined explicitly together with the derivation of the phase diagram and locations of critical curves three phases including a free or bulk phase an adsorbed phase and a ballistic or stretched phase are identified and the nature of the transitions between these phases are examined order parameters are defined and asymptotic expressions are determined for the critical adsorption surface as a function of the pulling force some of these results are also briefly generalized to arbitrary values of n

adsorbing motzkin paths the adsorption of a motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption the partition function of this model is determined exactly and the free energy is derived by examining the partition function the location of the critical point in the model is given exactly in one case and as the solution of a nonlinear equation in another case the phase diagram of the model is examined and i show that the transition is in general a first order adsorption transition except in cases where one or both endpoints of the path are restricted to be close to the adsorbing line

adsorbing and collapsing directed animals a model of a self interacting directed animal which also interacts with a solid wall is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition the directed animal is confined to a 45 wedge and it interacts with one of the walls of this wedge the existence of a thermodynamic limit in this model shown and the presence of an adsorption transition is demonstrated by using constructive techniques by comparing this model to a process of directed percolation we show that there is also a collapse or transition in this model we examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity the generating function of adsorbing directed animals in a half space is found next from which we find the tricritical exponents associated with the adsorption transition a full solution for a collapsing directed animal seems intractible so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next

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

directed compact percolation near a damp wall mean length and mean number of wall contacts key aspects of the cluster distribution in the case of directed compact percolation near a damp wall are derived as functions of the bulk occupation probability pand the wall occupation probability pw the mean length of finite clusters and mean number of contacts with the wall are derived exactly and we find that both results involve elliptic integrals and further multiple sum functions of two variables despite the added complication of these multiple sumfunctions ouranalysisshowsthatthecriticalbehaviourissimilartothedry wallcasewherethesefunctionsdonotappear wederivethecriticalamplitudes as a function of pw

pulled motzkin paths in this paper the models of pulled dyck paths in janse van rensburg 2010 j phys a math theor 43 215001 are generalized to pulled motzkin path models the generating functions of pulled motzkin paths are determined in terms of series over trinomial coefficients and the elastic response of a motzkin path pulled at its endpoint see orlandini and whittington 2004 j phys a math gen 37 5305 14 is shown to be r f 0 for forces pushing the endpoint toward the adsorbing line and r f f 1 2cosh f 2sinh f f as f for forces pulling the path away from the x axis in addition the elastic response of a motzkin path pulled at its midpoint is shown to be r f 0 for forces pushing the midpoint toward the adsorbing line and r f f 1 2cosh f 2 sinh f 2 2f as f for forces pulling the path away from the x axis formal combinatorial identities arising from pulled motzkin path models are also presented these identities are the generalization of combinatorial identities obtained in directed paths models to their natural trinomial counterparts

forces in motzkin paths in a wedge entropic forces in models of motzkin paths in a wedge geometry are considered as models of forces in polymers in confined geometries a motzkin path in the square lattice is a path from the origin to a point in the line y x while it never visits sites below this line and it is constrained to give unit length steps only in the north and east directions and steps of length in the north east direction motzkin path models may be generalized to ensembles of ne oriented paths above the line y rx where r 0 is an irrational number these are paths giving east north and north east steps from the origin in the square lattice and confined to the r wedge formed by the y axis and the line y rx the generating function gr of these paths is not known but if r 1 then i determine its radius of convergence to be and if r 0 1 then tr 1 3 the entropic force the path exerts on the line y rx may be computed from this an asymptotic expression for the force for large values of r is given by in terms of the vertex angle of the r wedge the moment of the force about the origin has leading terms as 0 and f 0 if 4 2 moreover numerical integration of the force shows that the total work done by closing the wedge is 1 085 07 lattice units an alternative ensemble of ne oriented paths may be defined by slightly changing the generating function gr in this model it is possible to determine closed form expressions for the limiting free energy and the force the leading term in an asymptotic expansions for this force agrees with the leading term in the asymptotic expansion of the above model and the subleading term only differs by a factor of 2

pulled directed lattice paths the generating functions of models of directed walks pulled by an external force f are determined using the kernel method these paths are models of linear polymers subject to an external force the generating function is related to the generating function of ballot paths and has an unexpected and non physical singularity for the model of dyck paths pulled at its central vertex in each model the force extension curve is determined exactly and has a sigmoid shape partially given by ctanh cf if f is the applied force and where c and c are the model dependent constants this result is consistent with previous results for models of pulled directed paths and also with data obtained from the numerical simulation of self avoiding walk models of linear polymers

mean length of finite clusters in directed compact percolation near a damp wall the mean length of finite clusters is derived exactly for the case of directed compact percolation near a damp wall we find that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case

forces in square lattice directed paths in a wedge a square lattice directed path confined to a wedge with vertex angle exerts an entropic force f on the wedge we show that this is a repulsive force of magnitude this force is determined by examining the combinatorial properties of the directed path and by determining the exact entropic contribution to the free energy in the limit as the path length goes to infinite

recovering risk neutral probability density functions from options prices using cubic splines and ensuring nonnegativity abstract we present a new approach to estimate the risk neutral probability density function pdf of the future prices of an underlying asset from the prices of options written on the asset the estimation is carried out in the space of cubic spline functions yielding appropriate smoothness the resulting optimization problem used to invert the data and determine the corresponding density function is a convex quadratic or semidefinite programming problem depending on the formulation both of these problems can be efficiently solved by numerical optimization software in the quadratic programming formulation the positivity of the risk neutral pdf is heuristically handled by posing linear inequality constraints at the spline nodes in the other approach this property of the risk neutral pdf is rigorously ensured by using a semidefinite programming characterization of nonnegativity for polynomial functions we tested our approach using data simulated from black scholes option prices and using market data for options on the s p 500 index the numerical results we present show the effectiveness of our methodology for estimating the risk neutral probability density function

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

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

manhattan lattice theta point exponents from kinetic growth walks and exact results from the nienhuis o n model kinetic growth walks kgw on the manhattan lattice have previously been shown to be equivalent to the static problem of interacting self avoiding walks on that lattice at the theta temperature here we illustrate how a complete set of exponents for the static problem including the crossover exponent phi and surface exponents at the ordinary and special transition points may be obtained from simulations of kinetic walks in the process we find that phi m 0 430 or 0 006 which encompasses the conjectured value 3 7 approximately 0 42857 for the theta point on an isotropic lattice our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition furthermore we point out that a recently examined variant of the nienhuis o n model can be mapped onto theta point walks on the manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures

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

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 directed walk model of a long chain polymer in a slit with attractive walls we present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential we consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit apart from the general interest in the effect of geometrical confinement this can be viewed as a two dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions we demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half plane geometry even when the polymer is constrained to be fixed at both ends on one wall we are not able to find a closed form solution for the free energy for finite width at all values of the interaction parameters but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane this allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur

a combinatorial interpretation of the free fermion condition of the six vertex model the free fermion condition of the six vertex model provides a five parameter sub manifold on which the bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple hence allowing explicit solutions such conditions arose originally in early field theoretic s matrix approaches here we provide a combinatorial explanation for the condition in terms of a generalized gessel viennot involution by doing so we extend the use of the gessel viennot theorem originally devised for non intersecting walks only to a special weighted type of intersecting walk and hence express the partition function of n such walks starting and finishing at fixed endpoints in terms of the single walk partition functions

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

an exact solution of two friendly interacting directed walks near a sticky wall we find the exact solution of two interacting friendly directed walks modelling polymers on the square lattice these walks are confined to the quarter plane by a horizontal attractive surface to capture the effects of dna denaturation and adsorption we find the solution to the model s corresponding generating function by means of the obstinate kernel method specifically we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts we then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants this expression guides our analysis of the model where we find the system exhibits four phases and proceed to delineate the full phase diagram showing that all observed phase transitions are second order

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

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

finite size scaling and universality for the totally asymmetric simple exclusion process the applicability of the concepts of finite size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one dimensional totally asymmetric simple exclusion process with open boundaries in the thermodynamic limit there are boundary induced transitions both of the first and second order between steady state phases of the model we derive finite size scaling expressions for the current near the continuous phase transition and for the local density near the first order transition under different stochastic dynamics and compare them to establish the existence of universal functions next we study numerically the finite size behavior of the lee yang zeros of the normalization factor for the different steady state probabilities

asymptotic lower bounds on hilbert numbers using canard cycles abstract in this work we give an asymptotic lower bound for the hilbert number for real planar polynomial differential systems this lower bound equals up to leading order to the best existing one but the method we provide is new and involves slow fast systems the construction strongly relies on generalized lienard systems

the raise and peel model of a fluctuating interface we propose a one dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter the adsorption rate at a special value of this parameter the model has some interesting features for example the spectrum is given by conformal field theory and the stationary non equilibrium probability distribution is given by the two dimensional equilibrium distribution of the ice model with domain wall type boundary conditions this connection is used to find exact analytic expressions for several quantities of the stochastic model vice versa some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model at the special point we study several properties of the model such as the height fluctuations as well as cluster and avalanche distributions the latter has a long tail which shows that the model exhibits self organized criticality we also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent 2 3 furthermore we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken