Towards a Combinatorial Understanding of Lattice Path Asymptotics

TL;DR

This paper introduces a new combinatorial approach to estimate the exponential growth of lattice path enumeration sequences, using comparisons with half-plane models, and discusses potential generalizations to higher dimensions.

Contribution

It presents a novel strategy for computing growth constants of lattice paths by comparing with half-plane models, providing tight bounds and combinatorial interpretations.

Findings

Bounds are often tight and match known formulas.

Provides a combinatorial interpretation of existing formulas.

Discusses potential for generalization to higher dimensions.

Abstract

We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the bounds are provably tight, and provide a combinatorial interpretation of recent formulas of Fayolle and Raschel (2012) and Bostan, Raschel and Salvy (2013). We discuss how to generalize to higher dimensions.