Counting walks with large steps in an orthant

Abstract

In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set $\mathcal{S}$ and confined to a given cone, especially the first quadrant of the plane, has been intensely studied. As a result, the generating functions of quadrant walks are now well-understood, provided the allowed steps are small, that is $\mathcal{S}\subset \{-1,0,1{\}}^2$. In particular, having small steps is crucial for the definition of a certain group of bi-rational transformations of the plane. It has been proved that this group is finite if and only if the corresponding generating function is D-finite (that is, it satisfies a linear differential equation with polynomial coefficients). This group is also the key to the uniform solution of 19 of the 23 small step models possessing a finite group.

In contrast, almost nothing is known for walks with arbitrary steps. In this paper, we extend the definition of the group, or rather of the associated orbit, to this general case, and generalize the above uniform solution of small step models. When this approach works, it invariably yields a D-finite generating function. We apply it to many quadrant problems, including some infinite families.

After developing the general theory, we consider the 13110 two-dimensional models with steps in $\{-2,-1,0,1{\}}^2$ having at least one $-2$ coordinate. We prove that only 240 of them have a finite orbit, and solve 231 of them with our method. The nine remaining models are the counterparts of the four models of the small step case that resist the uniform solution method (and which are known to have an algebraic generating function). We conjecture D-finiteness for their generating functions, but only two of them are likely to be algebraic. We also prove non-D-finiteness for the 12870 models with an infinite orbit, except for 16 of them.