Section: New Results

Random walks in orthants and lattice path combinatorics

Participant : Guy Fayolle.

In the second edition of the book [2], 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. 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 differential Galois theory, genus greater than 1 (i.e. when some jumps are of size $\ge 2$), etc. A recent topic 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 [25].