Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height

TL;DR

This paper studies self-avoiding walks on various lattices, providing rational generating functions for certain cases, constructing finite state machines for enumeration, and performing simulations to analyze properties of these walks.

Contribution

It introduces a method to compute rational generating functions for GSAWs on finite-height lattices and extends results to Greek key tours, resolving several conjectures.

Findings

Generating functions for GSAWs on half-infinite strips are rational.

Constructed finite state machines for enumeration of GSAWs.

Proved rationality of generating functions for Greek key tours.

Abstract

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.