A L Owczarek

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.

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.

Layering transitions for adsorbing polymers in poor solvents

An infinite hierarchy of layering transitions exists for model polymers in solution under poor solvent or low temperatures and near an attractive surface. A flat histogram stochastic growth algorithm known as FlatPERM has been used on a self- and surface interacting self-avoiding walk model for lengths up to 256. The associated phases exist as stable equilibria for large though not infinite length polymers and break the conjectured Surface Attached Globule phase into a series of phases where a polymer exists in specified layer close to a surface. We provide a scaling theory for these phases and the first-order transitions between them.

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.

A Class of Interaction-Round-a-Face Models and Its Equivalence with an Ice-Type Model

A new model (called the Temperley-Lieb interactions model) is introduced, in two-dimensional lattice statistics, on a square lattice ℒ. The Temperley-Lieb equivalence of this model to the six-vertex, self-dual Potts, critical hard-hexagons and critical nonintersecting string models is established. A graphical equivalence of this model to the six-vertex model generalizes this equivalence to noncritical cases of the above models. The order parameters of a specialization of this model are studied.

Exact Solution of the Discrete (1 + 1)-Dimensional SOS Model with Field and Surface Interactions

We present the solution of a linear solid-on-solid (SOS) model. Configurations are partially directed walks on a two-dimensional square lattice and we include a linear surface tension, a magnetic field, and surface interaction terms in the Hamiltonian. There is a wetting transition at zero field and, as expected, the behavior is similar to a continuous model solved previously. The solution is in terms ofq-series most closely related to theq-hypergeometric functions1φ1.

Stretching of a chain polymer adsorbed at a surface

In this paper we present simulations of a surface-adsorbed polymer subject to an elongation force. The polymer is modelled by a self-avoiding walk on a regular lattice. It is confined to a half-space by an adsorbing surface with attractions for every vertex of the walk visiting the surface, and the last vertex is pulled perpendicular to the surface by a force. Using the recently proposed flatPERM algorithm, we calculate the phase diagram for a vast range of temperatures and forces. The strength of this algorithm is that it computes the complete density of states from one single simulation. We simulate systems of sizes up to 256 steps.

Stacking Models of Vesicles and Compact Clusters

We investigate three simple lattice models of two dimensional vesicles. These models differ in their behavior from the universality class of partially convex polygons, which has been recently established. They do not have the tricritical scaling of those models, and furthermore display a surprising feature: their (perimeter) free energy is discontinuous with an isolated value at zero pressure. We give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq-series solutions and obtain the scaling functions where applicable.

Pulling absorbing and collapsing polymers from a surface

A self-interacting polymer with one end attached to a sticky surface has been studied by means of a flat-histogram stochastic growth algorithm known as FlatPERM. We examined the four-dimensional parameter space of the number of monomers (up to 91), self-attraction, surface-attraction and pulling force applied to one end of the polymer. Using this powerful algorithm the complete parameter space of interactions and pulling force has been considered. Recently it has been conjectured that a hierarchy of states appears at low-temperature/poor solvent conditions where a polymer exists in a finite number of layers close to a surface. We find re-entrant behaviour from the stretched phase into these layering phases when an appropriate force is applied to the polymer. Of interest is that the existence, and extent, of this re-entrant phase can be controlled not only by the force, but also by the ratio of surface-attraction to self-attraction. We also find that, contrary to what may be expected, the polymer desorbs from the surface when a sufficiently strong critical force is applied and does not transcend through either a series of de-layering transitions or monomer-by-monomer transitions. We discuss the problem mainly from the point of view of the stress ensemble. However, we make some comparisons with the strain ensemble, showing the broad agreement between the two ensembles while pointing out subtle differences.