S G Whittington

Interacting self-avoiding walks and polygons in three dimensions

Self-interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the -point. We consider self-avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon. We prove that these walks and polygons have the same limiting free energy if the interactions between nearest-neighbour vertices are repulsive. The attractive interaction regime is investigated using Monte Carlo methods, and we find evidence that the limiting free energies are also equal here. In particular, this means that these models have the same -point, in the asymptotic limit. The dimensions and shapes of walks and polygons are also examined as a function of the interaction strength.

Adsorption of a directed polymer subject to an elongational force

We consider several different directed walk models of a homopolymer adsorbing at a surface when the polymer is subject to an elongational force which hinders the adsorption. We use combinatorial methods for analyzing how the critical temperature for adsorption depends on the magnitude of the applied force and show that the crossover exponent φ changes when a force is applied. We discuss the characteristics of the model needed to obtain a re-entrant phase diagram.

Monte Carlo study of the percolating cluster for the square lattice site problem

Clusters which just span finite lattices of various sizes have been generated using a Monte Carlo method. These have been analysed to form estimates of the mean values of cyclomatic index, valence and perimeter. In addition the shortest spanning self-avoiding walk has been characterised. The ramified nature of these clusters is discussed in terms of these properties.

On the behaviour of collapsing linear and branched polymers

In this paper, we have reviewed the behaviour of a variety of models of collapse transitions in linear and branched polymers. Both rigorous and numerical results are presented. Some of the work described is still in progress, so the results are incomplete. Nevertheless, the overall picture demonstrates how the association of an attractive fugacity between nearby monomers can give rise to a collapse transition in a wide variety of models. These then qualitatively describe the transitions undergone by collapsing linear and branched polymers.

Extension of a theorem on super-multiplicative functions

Concerns a generalisation of the fundamental theorem on super-multiplicative functions for which the super-multiplicative inequality is replaced by an+m>or=anaf(m) with limm to infinity m-1f(m)=1.

Critical exponents for lattice animals with fixed cyclomatic index

The authors derive an inequality between the number of trees and the number of lattice animals with exactly c cycles, an(c), for all positive c. If they assume that an(c) approximately n- theta c lambda nc, n to infinity , c fixed, they use this to show that theta c= theta o-c where theta 0 is the corresponding exponent for trees.

Random linking of lattice polygons

We study the linking probability of polygons on the simple cubic lattice. In particular, we consider two polygons each having n edges, confined to a cube of side L, and ask for the linking probability as a function of n and L. We also consider other situations in which the polygons are restricted to be not too far apart, but not necessarily confined to a cube. We prove several rigorous results, and use Monte Carlo methods to address some questions which we are unable to answer rigorously. An interesting feature is that the linking probability is a function of L/nv, where v is the exponent characterizing the radius of gyration of a polygon.

Self-Avoiding Walks with Geometrical Constraints

We consider self-avoiding walks on aD-dimensional hypercubic lattice, confined to a slab geometry and confined to a half-space. We present a proof of the existence of a “connective constant” for the slab geometry and review some corresponding results for the half-space. We also discuss the way in which scaling arguments can be used to give stronger, but nonrigorous, results.

Branched polymers with a prescribed number of cycles

The authors consider the number of lattice animals with exactly c cycles. For weakly embeddable (bond) clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c. They use series analysis methods to estimate the critical exponents for c=1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles. Evidence from series analysis results and from expansions in inverse powers of the dimension suggests that, in the case of strong embeddings (i.e. site clusters) the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings (i.e. bond clusters). They discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity.

A self-avoiding walk model of random copolymer adsorption

There is an error in the proof of lemma 3.4 which can be remedied by using (3.22) and (3.13) directly. The proof of lemma 3.2 can also be simplified. Please see PDF for details.