Counting walks with large steps in an orthant

TL;DR

This paper extends the study of lattice walks in the first quadrant to larger step sets, developing a generalized group approach that identifies when generating functions are D-finite and solving many complex models.

Contribution

It generalizes the group method for lattice walks to arbitrary step sets, enabling the solution of many previously intractable models and identifying conditions for D-finiteness.

Findings

240 models have finite orbits, 231 solved with the new method

9 models resist the uniform solution, likely algebraic

Most models with infinite orbits are proven non-D-finite

Abstract

In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions ofquadrant walks are now well-understood, provided the allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular, having smallsteps is crucial for the definition of a certain group of bi-rationaltransformations of the plane. It has been proved that this group is finite ifand only if the corresponding generating function is D-finite (that is, it satisfies a lineardifferential equation with polynomial coefficients). This group is also thekey to the uniform solution of 19 of the 23 small step models possessing afinite group.In contrast, almost nothing is known for walks with arbitrary steps. In thispaper, we extend the definition of the group, or rather of the associatedorbit, to this general case, and generalize the above uniform solution ofsmall step models. When this approach works, it invariably yields a D-finitegenerating function. We apply it to many quadrant problems, including some infinite families.After developing the general theory, we consider the $13\ 110$ two-dimensionalmodels with steps in $\{-2,-1,0,1\}^2$ having at least one $-2$ coordinate. Weprove that only 240 of them have a finite orbit, and solve 231 of them withour method. The 9 remaining models are the counterparts of the 4 models of thesmall step case that resist the uniform solution method (and which are knownto have an algebraic generating function). We conjecture D-finiteness for their generatingfunctions, but only two of them are likely to be algebraic. We also provenon-D-finiteness for the $12\ 870$ models with an infinite orbit, except for16 of them.