Abstract
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey 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.
Highlights
Over the last two decades there has been a flurry of activity regarding lattice walks restricted to certain types of steps and to certain subsets of the lattice
(In the second case the periodicity will not be apparent in the series for the full plane, but will be visible for the half plane.) We introduce another new definition: a two-step rule is aperiodic if there exists a k 1 such that for any i, j ∈ {east, north, west, south}, (Tk)ij > 0
We have investigated walks on the square lattice Z2 obeying two-step rules, which govern which consecutive pairs of steps in the four cardinal directions are permitted
Summary
Over the last two decades there has been a flurry of activity regarding lattice walks restricted to certain types of steps and to certain subsets of the lattice. Bousquet-Melou and Mishna [6] showed that there are 79 non-isomorphic models with a step set S ⊂ {−1, 0, 1}2 \ {(0, 0)} (so-called small steps, where walks can step along the edges of the square lattice or take diagonal steps across squares), and conjectured that 23 of those have holonomic (or differentially finite, usually written Dfinite for short) generating functions, four of which are algebraic. They derived solutions for 22 of the 23 holonomic cases. In at least some cases the methods used in the aforementioned works can be applied here, but there are still many techniques which we have not yet attempted to use
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