Self-avoiding walks and polygons confined to a square

Abstract

We prove several rigorous results about the asymptotic behaviour of the numbers of polygons and self-avoiding walks confined to a square on the square lattice. Specifically we prove that the dominant asymptotic behaviour of polygons confined to an L × L square is identical to that of self-avoiding walks that cross an L × L square from one corner vertex to the opposite corner vertex. We prove results about the sub-dominant asymptotic behaviour of self-avoiding walks crossing a square and polygons confined to a square and extend some results to self-avoiding walks and polygons in a hypercube in .