Walks obeying two-step rules on the square lattice: full, half and quarter planes

TL;DR

This paper studies two-step walk rules on the square lattice, classifying their properties and analyzing their generating functions, asymptotics, and limiting shapes in various lattice regions, including initial exploration of quarter plane models with novel algebraic and D-finite solutions.

Contribution

It introduces a classification of two-step walk rules on the square lattice and explores their enumerative and algebraic properties, including group structures and solution methods, extending beyond regular short-step models.

Findings

Some models have finite groups with D-finite generating functions.

Others have infinite groups with algebraic or non-D-finite solutions.

New phenomena include models with algebraic or D-finite functions despite infinite groups.

Abstract

We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a number of criteria, and show how these properties affect their generating functions, asymptotic enumerations and limiting shapes, on the full lattice as well as the upper half plane. For walks in the quarter plane, we only make a few tentative first steps. We propose candidates for the group of a model, analogous to the group of a regular short-step quarter plane model, and investigate which models have finite versus infinite groups. We demonstrate that the orbit sum method used to solve a number of the original models can be made to work for some models here, producing a D-finite solution. We also generate short series for all models and guess differential or algebraic equations where possible. In doing so, we find that there are possibilities here which do not occur for the regular short-step models, including cases with algebraic or D-finite generating functions but infinite groups, as well as models with non-D-finite generating functions but finite groups.