[Paper Review] Inhomogeneous Restricted Lattice Walks

Manfred Buchacher, Manuel Kauers|arXiv (Cornell University)|Nov 16, 2018

Advanced Combinatorial Mathematics15 references4 citations

TL;DR

This paper studies inhomogeneous restricted lattice walks in half-spaces and the quarter plane, where step sets depend on position or time. It proves that generating functions for inhomogeneous half-space walks are always algebraic using a generalized kernel method for linear systems, and presents an experimental classification of short-step quarter-plane models, revealing many D-finite cases without a theoretical explanation yet.

ABSTRACT

We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. For the models in the quarter plane, we have carried out an experimental classification of all models with small steps. We discovered many (apparently) D-finite cases for most of which we have no explanation yet.

Motivation & Objective

• To systematically analyze inhomogeneous lattice walk models where step sets vary with position or time, extending classical results on homogeneous models.
• To generalize the kernel method to linear systems of functional equations for inhomogeneous half-space walks.
• To classify short-step inhomogeneous models in the quarter plane and investigate the nature of their generating functions.
• To identify patterns in D-finiteness for inhomogeneous models, especially where no theoretical explanation exists.

Proposed method

• Generalizes the kernel method to solve linear systems of functional equations arising from inhomogeneous half-space walks.
• Uses polynomial inhomogeneities defined by linear functions in position and step count, with periodic switching between step sets.
• Applies group action and orbit sums (via the group generated by (x,y)↦(x,1/y) and (x,y)↦(1/x,y)) to eliminate boundary terms in functional equations.
• Employs formal power series and Laurent polynomial representations of step sets to derive closed-form expressions for generating functions.
• Uses experimental classification via computer algebra to analyze models with small steps and two inhomogeneity patterns in the quarter plane.
• Applies D-finite closure properties to infer D-finiteness of full generating functions from partial solutions.

Experimental results

Research questions

• RQ1 Are generating functions for inhomogeneous lattice walks in a half-space always algebraic, even when step sets depend on position and time?
• RQ2 Can the kernel method be extended to handle linear systems of functional equations in inhomogeneous models?
• RQ3 Why do many inhomogeneous quarter-plane models with small steps yield D-finite generating functions despite lacking a theoretical explanation?
• RQ4 What structural properties of inhomogeneous models lead to D-finite or algebraic generating functions?
• RQ5 How can the dimension concept from homogeneous models be generalized to inhomogeneous inhomogeneous lattice walks?

Key findings

• The generating function for inhomogeneous lattice walks in a half-space is always algebraic, generalizing the classical result of Banderier and Flajolet.
• A generalized kernel method for linear systems of functional equations successfully proves algebraicity in the half-space case.
• For inhomogeneous quarter-plane models with small steps, a large number of generating functions are found to be D-finite, though no explanation is currently available.
• The functional equations for quarter-plane models are solved using orbit sums under a group action, which eliminates boundary terms and enables extraction of positive parts.
• The full generating function F(x,y) = F₀(x,y) + F₁(x,y) is D-finite due to closure properties and the derived expression involving group orbit sums.
• The experimental classification reveals that D-finiteness is prevalent in inhomogeneous models, suggesting deeper structural patterns yet to be uncovered.