Section: New Results

Random Walks in Orthants

Participant : Guy Fayolle.

The Second Edition of the Book [45] Random walks in the Quarter Plane, prepared in collaboration with R. Iasnogorodski (St-Petersburg, Russia) and V. Malyshev (MGU, Moscow), is complete and now in the Springer Production Department. It will be published in the collection Probability Theory and Stochastic Processes. Part II of this second edition borrows specific case-studies from queuing theory, and enumerative combinatorics. Five chapters have been added, including examples and applications of the general theory to enumerative combinatorics. Among them:

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

• Chapter Coupled-Queues shows the first example of a queuing 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.