On the functions counting walks with small steps in the quarter plane

TL;DR

This paper analyzes the generating functions of quarter-plane walks with small steps, revealing their complex multi-valued nature and proving a conjecture about their non-holonomy for certain models.

Contribution

It characterizes the analytic structure of generating functions for all non-singular small-step walks in the quarter plane, proving non-holonomy in specific cases and confirming a conjecture.

Findings

Functions have infinitely many meromorphic branches with identified poles.

For 51 models, the generating functions are holonomic on one dense subset of z-values and non-holonomic on the other.

The overall generating function is non-holonomic, confirming a prior conjecture.

Abstract

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\'elou and Mishna.