Compter les marches dans le quadrant par les invariants de Tutte & théorie des transformations des fonctions elliptiques
salle 435 (UMPA)
In the 70's, Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly coloured triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. To be applicable, the method requires the existence of two functions called "invariant", and "decoupling function", respectively. We construct those using the interpretation of the kernel of the model as a Riemann surface of genus 1, and using the transformation theory of elliptic functions.