[en] 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. (author)

[en] We state the diffusion algebra equations of the stationary state of the three parameter (α, β and q) asymmetric simple exclusion Process as a linear functional $\mathcal{L},$ acting on a tensor algebra. From $\mathcal{L}$ we construct a pair of sequences, $\{P_n\}$ and $\{Q_m\}$, of monic polynomials which are bi-orthogonal, that is, they satisfy $\mathcal{L}(P_n{Q}_m)={\mathrm{\Lambda <!--\; ??\; -->}}_n{\mathrm{\delta <!--\; ??\; -->}}_{n,m}$(where ${\mathrm{\Lambda <!--\; ??\; -->}}_n$ is a scalar). The uniqueness and existence of the pair of sequences arises from the determinant of the bi-moment matrix whose elements satisfy a pair of q-recurrence relations. The determinant is evaluated using an LDU-decomposition. If the linear functional is represented as an inner product, $\mathcal{L}(\cdot <!--\; ???\; -->)=⟨<!--\; ???\; -->W|\cdot <!--\; ???\; -->|V⟩<!--\; ???\; -->$ then the action of the polynomials Q_n on the boundary vector $|V⟩<!--\; ???\; -->$ generate a basis $|V_n⟩<!--\; ???\; -->=Q_n|V⟩<!--\; ???\; -->$ whose orthogonal dual vectors are given by the action of P_n on the dual boundary vector $⟨<!--\; ???\; -->W|$, that is $⟨<!--\; ???\; -->W_n|=⟨<!--\; ???\; -->W|P_n$. This basis gives the representation of the algebra which is associated with the Al-Salam–Chihara polynomials obtained by Sasamoto. (paper)

[en] We present results for the generating functions of polygons and more general objects that can touch, constructed from two fully directed walks on the infinite triangular lattice, enumerated according to each type of step and weighted proportional to the area and the number of contacts between the directed sides of the objects. In general these directed objects are known as festoons, being constructed from the so-called friendly directed walks, while the subset constructed from vicious walks are staircase polygons, also known as parallelogram polyominoes. Additionally, we give explicit formulae for various first area-moment generating functions, that is when the area is summed over all configurations with a given perimeter. These results generalize and summarize nearly all known results on the square lattice, since such results can be obtained by setting one of the step weights to zero. All our results for the triangular lattice are new and hence provide the opportunity to study subtle changes in scaling between lattices. In most cases we give our results both in terms of ratios of infinite q-series and as continued fractions. (author)

[en] We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary 'decorated' weights as well as an arbitrary 'background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence, we also present an efficient method for finding explicit closed-form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed-form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as the models of steric stabilization and sensitized flocculation.

[en] 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=0 to 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_t(y;k)=Σ_m≥0r-'^W_t(y;m)k^m where r-'^W_t(y;m) is the number of n-path configurations of watermelon type having deviation γ for which the path closest to the surface returns to the surface m times. The 'marked return' polynomial is defined by u-'^W_t(y;k₁)≡R-'^W₁(y;k₁+l)=Σ_m≥0u-'^W_t(y;m)k₁^m) where u-'^W_t(y;m) is the number of marked configurations having at least m returns, just m of which are marked. Both r-'^W_t(y;m) and u-'^W(y;m) are expressed in terms of the numbers of paths ignoring returns but introducing a suitably modified endpoint condition. This enables u-'^W_t(y;m) to be written in product form for arbitrary y, but for r-'^W_t(y;m) this can only be done in the case y=0. (author)

[en] We present the exact solution of a three-dimensional lattice model of a polymer confined between two sticky walls, that is within a slab. We demonstrate that the model behaves in a similar way to its two-dimensional analogues and agrees with Monte Carlo evidence based upon simulations of self-avoiding walks in slabs. The model on which we focus is a variant of the partially directed walk model on the cubic lattice. We consider both the phase diagram of relatively long polymers in a macroscopic slab and the effective force of the polymer on the walls of the slab.

[en] We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday. (paper)

[en] We show that the known matrix representations of the stationary state algebra of the asymmetric simple exclusion process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the constant term method (CTM) and bijective combinatorial methods to express many forms of the ASEP normalization factor in terms of ballot numbers. One particular lattice path representation shows that the coefficients in the recurrence relation for the ASEP correlation functions are also ballot numbers. Additionally, the CTM has a strong combinatorial connection which leads to a new 'canonical' lattice path representation and to the 'ο-expansion' which provides a uniform approach to computing the asymptotic behaviour in the various phases of the ASEP. The path representations enable the ASEP normalization factor to be seen as the partition function of a more general polymer chain model having a two-parameter interaction with a surface. We show, in the case α β = 1, that the probability of finding a given number of particles in the stationary state can be expressed via non-intersecting lattice paths and hence as a simple determinant

[en] We consider Motzkin path models for polymers confined to a slit. The path interacts with each of the two confining lines, and we define parameters a and b to characterize the strengths of the interactions with the two lines. We consider the cases where (i) each vertex in a confining line contributes to the energy and (ii) where each edge in a confining line contributes to the energy. For the vertex and edge versions of Motzkin paths, we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the (a, b)-plane, and asymptotically (i.e. for large slit widths) elsewhere. We find regions where the force between the lines is long range and repulsive, short range and repulsive and short range and attractive. Our results indicate that the general form of the phase diagram is model independent, and similar to previous results for a Dyck path model, although the details do depend on the underlying configurational model. We also contrast the method used here to find the generating function with the transfer matrix and heap methods

[en] 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