Figure eights on the square lattice: enumeration and Monte Carlo estimation

This paper concerns the numbers of figure eights weakly embeddable in a lattice, important in the theories of self-avoiding walks and the Ising problem. Two Monte Carlo techniques for estimating such numbers are outlined and have been applied to the case of the square lattice. Enumerations have also been carried out for this lattice, and extrapolation formulae have been sought on the basis of these results. It appears that, as one goes to large graphs having a given number of edges, the numbers of figure eights will become comparable to those of self-avoiding polygons, and may be larger. Some rigorous, but rather weak, bounds are established for the numbers of such figure eights.