Self-avoiding walks which cross a square

Abstract

The authors consider self-avoiding walks on the square lattice which are confined to lie in or on the boundary of a square with vertices at (0, 0), (0, L), (L, 0) and (L, L). They ask for the number of such walks which begin at the origin and end at the vertex (L, L), especially in the large L limit. Similarly they ask for the mean number of steps in such walks as a function of L. At fixed L the authors also associate a fugacity with the number of steps of the walk and ask how the system behaves as a function of this fugacity. They provide some rigorous results, in particular proving that there is a phase transition at some particular value of the fugacity, and supplement these with the anal..