Self-avoiding walks and polygons confined to a square

TL;DR

This paper rigorously analyzes the asymptotic behavior of self-avoiding walks and polygons confined to squares on a lattice, revealing their dominant asymptotics are identical and extending results to higher dimensions.

Contribution

It establishes the equivalence of asymptotic behaviors of polygons and crossing walks in squares and extends these results to hypercubes in higher dimensions.

Findings

Polygons and crossing walks share the same dominant asymptotic behavior.

Results are extended to self-avoiding walks and polygons in hypercubes.

New insights into the subdominant asymptotics of these models.

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 LxL square is identical to that of self-avoiding walks that cross an LxL 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 polygons confined to a square. In addition, we investigate the problems of self-avoiding walks and polygons in a hypercube in the d-dimensional hypercubic lattice.