Combinatorics of lattice paths and polygons

© 2012 Dr. Nicholas Ross BeatonWe consider the enumeration of self-avoiding walks and polygons on regular lattices. Such objects are connected with many other problems in combinatorics, as well as in fields as diverse as physics and chemistry. We examine the general models of walks and polygons and methods we can use to study them; subclasses whose properties enable a rather deeper analysis; and extensions of these models which allow us to model physical phenomena like polymer collapse and adsorption. While the general models of self-avoiding walks and polygons are certainly not considered to be ‘solved’, recently a great deal of progress has been made in developing new methods for studying these objects and proving rigorous results about their enumerative properties, particularly on the honeycomb lattice. We consider these recent results and show that in some cases they can be extended or generalised so as to enable further proofs, conjectures and estimates. In particular, we find that properties shared by all two-dimensional lattices allow us to develop new methods for estimating the growth constants and certain amplitudes for the square and triangular lattices. The subclasses of self-avoiding walks and polygons that we consider are typically defined by imposing restrictions on the way in which a walk or polygon can be constructed. Ideally the restrictions should be as weak as possible, so as to result in a model that closely resembles the unrestricted case, while still enabling some manner of simple recursive construction. These recursions can sometimes lead to solutions for generating functions or other quantities of interest. Some of the models we consider display quite unusual asymptotic properties despite the relatively simple restrictions which lead to their construction. We approach the modelling of polymer adsorption in several ways. Firstly, we adapt some of the new methods for studying self-avoiding walks on the honeycomb lattice to account for interactions with an impenetrable surface. In this way we are able to prove the exact value of the critical surface fugacity for adsorbing walks, confirming an existing conjecture. Then, we show that some key identities for the honeycomb lattice model lead to a new method for estimating the critical surface fugacities for adsorption models on the square and triangular lattices. Many of the estimates we obtain in this way are new; for the cases where previous estimates did already exist, our results are several orders of magnitude more precise. Finally, we define some new solvable models of polymer adsorption which generalise existing models, and find that some of these models exhibit interesting and unexpected critical behaviour