Section: New Results

Random Walks in Orthants

Participant : Guy Fayolle.

In the book [3] , original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, 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 in [3] for the solution to be rational or algebraic. When the underlying algebraic curve is of genus 1, we propose, in collaboration with R. Iasnogorodski (St-Petersburg, Russia), a concrete criterion ensuring the finiteness of the group. It turns out that this criterion is always tantamount to the cancellation of a single constant, which can be expressed as the determinant of a matrix of order 3 or 4, and depends in a polynomial way on the coefficients of the walk [20] .

In collaboration with R. Iasnogorodski (St-Petersburg, Russia) and V. Malyshev, we prepared the second edition of the book [3] , which will be published by Springer, in the collection Probability Theory and Stochastic Processes. Part II of this second edition borrows specific case-studies from queueing theory, and enumerative combinatorics. Five chapters will be added, including examples and applications of the general theory to enumerative combinatorics. Among them:

• Explicit criterions for the finiteness of the group, both in the genus 0 and genus 1 cases.

• Chapter Coupled-Queues shows the first example of a queueing system analyzed by reduction to a BVP in the complex plane.

• Chapter Joining the shorter-queue analyzes a famous model, where maximal homogeneity conditions do not hold, hence leading to a system of functional equations.

• Chapter Counting Lattice Walks concerns the so-called enumerative combinatorics. When counting random walks with small steps, the nature (rational, algebraic or holonomic) of the generating functions can be found and a precise classification is given for the basic (up to symmetries) 79 possible walks.