New results on the enumeration of nonintersecting random walks (Q1095920)

Many attempts have been made to obtain approximations and closed forms for the generating function enumerating non-intersecting (self-avoiding) walks on a periodic lattice. In this paper we introduce an approach based on the Mayer cluster theory of imperfect gases and liquids and apply it to a continuum model and to a lattice model. The combinatorics are appreciably simpler than for the Mayer problem. We have not yet solved any such problems analytically, but a combination of analytical work and a series expansion enables us to approach the exact generating functions as closely as we please. The starting point is an approximation very similar to the Percus-Yevick approximation for liquids and this can be pregressively improved upon.