Section: New Results

Random Walks in Orthants

Participant : Guy Fayolle.

The Second Edition of the Book [39] Random walks in the Quarter Plane, prepared in collaboration with R. Iasnogorodski (St-Petersburg, Russia) and V. Malyshev (MGU, Moscow), has been 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 have been added, including examples and applications of the general theory to enumerative combinatorics. Among them:

• Explicit criteria 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.