Fast construction of self-avoiding polygons and efficient evaluation of closed walk fractions on the square lattice

Abstract

We build upon a recent theoretical breakthrough by employing novel algorithms to accurately compute the fractions $F_p$ of all closed walks on the infinite square lattice whose the last erased loop corresponds is any one of the $762, 207, 869, 373$ self-avoiding polygons $p$ of length at most 38. Prior to this work, only 6 values of $F_p$ had been calculated in the literature. The main computational engine uses efficient algorithms for both the construction of self-avoiding polygons and the precise evaluation of the lattice Green's function. Based on our results, we propose two conjectures: one regarding the asymptotic behavior of sums of $F_p$, and another concerning the value of $F_p$ when $p$ is a large square. We provide strong theoretical arguments supporting the second conjecture. Furthermore, the algorithms we introduce are not limited to the square lattice and can, in principle, be extended to any vertex-transitive infinite lattice. In establishing this extension, we resolve two open questions related to the triangular lattice Green's function.