On the small-step quarter plane lattice walks with a non D-finite univariate generating function

TL;DR

This paper investigates the D-finiteness of univariate generating functions for small-step quarter-plane lattice walks, providing partial proofs, conjectures, and identifying models with differentially algebraic properties.

Contribution

It advances understanding of the conjecture relating walk group finiteness to D-finiteness, offering proofs for specific models and proposing new conjectures.

Findings

Identified 21 models with non-D-finite generating functions.

Proved some models have differentially algebraic endpoint series.

Numerical estimates suggest non-D-finiteness in several boundary series.

Abstract

We report on the status of the conjecture of Bousquet-M\'elou and Mishna that the univariate counting generating function of a small-step quarter-plane lattice model is D-finite if and only if the group of the walk is finite. While the finite-group case is fully resolved, the infinite-group case remains incomplete. We list the arguments for the non-D-finiteness for 21 of the 56 infinite-group models: the five singular models, three models with zero drift and thirteen models with polar interior drift. The proof of the latter two families uses asymptotic results of Bostan--Raschel--Salvy combined with probabilistic estimates of Denisov--Wachtel and Duraj. We further identify nine infinite-group models whose endpoint counting series are differentially algebraic via decoupling functions, though this does not settle their D-finiteness. For 21 of the remaining models, numerical estimation of singular exponents suggests non-D-finiteness of one of its the boundary series $Q(1,0;t)$ or $Q(0,1;t)$, and we state a conjecture to this effect.