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

Kilian Raschel (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

The aim of this article is to introduce a unified method for giving explicit integral representations of the trivariate generating function of the number of paths for walks with small steps confined in a quadrant. For a number of such walks, this yields for the first time an explicit expression of this counting function. Moreover, the nature of the integrand of the integral formulations obtained here is shown to be very directly dependent on the finiteness of a well-known and naturally attached group of birational transformations.

Mots clés: uniformization; lattice walks; generating function; Riemann-Carleman boundary value problem; conformal gluing function; Weierstrass elliptic function; Riemann surface; uniformization.

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Kilian Raschel. Counting walks in a quadrant : a unified approach via boundary value problems. 2010. ⟨hal-00461853v2⟩