Random self-avoiding walks on one-dimensional lattices

Abstract

For a general class of one-dimensional lattices, we show that the generating function for self-avoiding walks can be explicitly expressed in terms of a generating matrix. Further, the connective constant can be determined by calculating eigenvalues of the generating matrix. This matrix is also used to get asymptotic results for random self-avoiding walks, in particular a Central Limit Theorem for the endpoint. The asymptotic results also show that it is possible to define an infinite random self-avoiding walk. Numerical calculations are performed for strips in the plane square lattice and some other one-dimensional lattices. The results may be extended to self-avoiding trails and to self-avoiding random walks.