An elementary solution of Gessel's walks in the quadrant

TL;DR

This paper provides an elementary, constructive proof for Gessel's conjecture on lattice walks in the quadrant, simplifying previous complex algebraic and analytical methods and extending to other models.

Contribution

It introduces a new elementary proof technique for Gessel's walks, previously proven only through complex algebraic and analytical methods, and applies this approach to other algebraic models.

Findings

Elementary proof of Gessel's conjecture achieved

Extension of method to other algebraic quadrant models

Simplification of previous complex algebraic proofs

Abstract

Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an elementarysolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.