## Section: New Results

### Lattice path combinatorics

Participant : Guy Fayolle.

In the second edition of the book [39], original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps (size 1), the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called *group of the walk*, allows to deduce theoretical features about the nature of the solutions. In particular, when the *order* of the group is finite, necessary and sufficient conditions have been given for the solution to be rational, algebraic or $D$-finite (i.e. solution of a linear differential equation) in which case the underlying algebraic curve is of genus 0 or 1. In this framework, number of difficult open problems related to lattice path combinatorics are currently being explored, in collaboration with A. Bostan and F. Chyzak (project-team SPECFUN, Inria-Saclay), both from theoretical and computer algebra points of view: concrete computation of the criteria, utilization of Galois theory for genus greater than 1 (i.e. when some jumps are $\ge 2$), etc.