Singularity analysis via the iterated kernel method

Abstract

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach of Fayolle, Rachel, and Kurkova. Here we consider these five models, called the singular models, and prove that the generating functions marking the number of walks of a given length are not D-finite -- thus finishing the proof of a conjecture of Bousquet-Mélou and Mishna. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks.

23 pages, added new introduction, other clarifications and fixes

References in corpus (1)

• On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

Cited by in corpus (25)

• Intervals in the greedy Tamari posets
• Counting quadrant walks via Tutte's invariant method
• On 3-dimensional lattice walks confined to the positive octant
• On the nature of the generating series of walks in the quarter plane
• An elementary solution of Gessel's walks in the quadrant
• Square lattice walks avoiding a quadrant
• Weighted Lattice Walks and Universality Classes
• Walks in the quarter plane, genus zero case
• New steps in walks with small steps in the quarter plane
• Hopf dreams and diagonal harmonics
• Length derivative of the generating series of walks confined in the quarter plane
• Enumeration of three-quadrant walks via invariants: some diagonally symmetric models
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Continued Classification of 3D Lattice Walks in the Positive Octant
• On the Kernel curves associated with walks in the quarter plane
• Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks
• On the nature of four models of symmetric walks avoiding a quadrant
• Non-D-finite excursions in the quarter plane
• Asymptotic lattice path enumeration using diagonals
• Infinite Orders and Non-$D$-finite Property of $3$-Dimensional Lattice Walks
• Towards a Combinatorial Understanding of Lattice Path Asymptotics
• Walks obeying two-step rules on the square lattice: full, half and quarter planes
• Walks avoiding a quadrant and the reflection principle
• Lattice Walks in the Octant with Infinite Associated Groups
• Walks in the Quarter Plane with Multiple Steps