Counting walks in a quadrant: a unified approach via boundary value problems

TL;DR

This paper introduces a unified boundary value problem approach to derive explicit integral formulas for counting walks confined to a quarter plane, revealing new expressions for the generating functions of various models.

Contribution

It provides a novel, unified method to obtain explicit integral representations of generating functions for quarter-plane walks, linking their form to group finiteness and walk covariance.

Findings

Explicit integral formulas obtained for many models

The integrand's nature depends on group finiteness

The approach unifies previous disparate methods

Abstract

The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk