Lattice Polygons and Related Objects

Self-avoiding lattice polygons, i.e. embeddings of simple closed curves in a lattice, have been studied for more than fifty years. They are interesting as models of ring polymers in solution in good solvents and they appear in graphical expansions in, for instance, the Ising problem. They have been counted exactly (see e.g. [17]) and studied by Monte Carlo methods (see e.g. [22]). We understand many of their prop-erties but rigorous results are scarce. In 1961 Hammersley [11] showed that they grow exponentially at the same rate as self-avoiding walks—a result which was by no means obvious at the time—but we still know very little about the sub-dominant asymptotic behaviour, except in dimensions higher than four where lace expansion techniques are useful.

This chapter will review some of the results which have been established rigor-ously. Apart from results about the numbers of polygons we also discuss counting polygons by both perimeter and area, which gives an interesting model of vesicles and how they respond to an osmotic force. This is closely related to the problem of self-avoiding surfaces. In three dimensions polygons can be knotted, and we in-vestigate what is known rigorously about knot probabilities. There are many open questions in this area and we mention some of these. We then turn to polygons with a geometrical constraint and consider polygons confined to a wedge or to a slit or slab. The main interest has focussed on whether the constraint changes the exponential growth rate of the number of polygons. Finally we give a brief account of some results on lattice trees and lattice animals.