New steps in walks with small steps in the quarter plane

Irina Kourkova (1) , Kilian Raschel (2)

1 Modélisation stochastique
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

In this article we present a new approach for finding the generating function counting (non-singular) walks with small steps in the quarter plane. We obtain its explicit expression in terms of infinite series, while in the literature, its standard expression is given in terms of solutions to boundary value problems. As an illustration of the computational interest of our approach, we work out these new formulae for Kreweras' model: this leads to an expression for the generating function with Weierstrass Zeta functions, and eventually (using transformation theory of elliptic functions) to the well-known algebraic expression.

Mots clés: Walks in the quarter plane; Counting generating function; Holonomy; Group of the walk; Riemann surface; Elliptic functions; Uniformization; Universal covering

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

Irina Kourkova, Kilian Raschel. New steps in walks with small steps in the quarter plane. 2013. ⟨hal-00840239v1⟩