Explicit expression of the counting generating function for Gessel's walk

Irina Kurkova (1) , Kilian Raschel (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider the so-called Gessel's walk, that is the planar random walk that is confined to the first quadrant and that can move in unit steps to the West, North-East, East and South-West. For this walk we make explicit the generating function of the number of paths starting at $(0,0)$ and ending at $(i,j)$ in time $k$.

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

Domaines: Combinatoire [math.CO]

Irina Kurkova, Kilian Raschel. Explicit expression of the counting generating function for Gessel's walk. 2009. ⟨hal-00438190v1⟩