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Self-avoiding walks and polygons confined to a square

We prove two 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 also prove a result about the subdominant asymptotic behaviour of self-avoiding walks crossing a square and extend this result to self-avoiding walks crossing a hypercube in Z d .