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## Section: New Results

### Random walks in orthants and lattice path combinatorics

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 and the underlying algebraic curve is of genus 0 or 1, 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 this framework, a number of difficult open problems related to lattice-path combinatorics are currently being explored by Alin Bostan, Frédéric Chyzak, and Guy Fayolle, both from the theoretical and computer-algebra viewpoints: concrete computation of the criteria, utilization of differential Galois theory, genus greater than 1 (i.e., when some jumps are of size $\ge 2$), etc. A recent topic of future research deals with the connections between simple product-form stochastic networks (so-called Jackson networks) and explicit solutions of functional equations for counting lattice walks, see [17].