Andrew Elvey Price

Given a set of small steps, we consider the three variable generating function counting walks in the quarter plane using these steps. Since the seminal paper by Bousquet-Mélou and Mishna, the problem of characterising the generating function into the hierarchy Algebraic ⊂ D-finite ⊂ D-algebraic has received a lot of attention. For unweighted walks this characterisation was completed in 2018, however the existing proof of D-finiteness does not generalise to weighted walks. In this talk I will describe our recent proof that the generating function is D-finite in each variable if and only if the group of the walk is finite. This result applies to any weighted model and is based on the elliptic function method. 
This is joint work with Thomas Dreyfus and Kilian Raschel.