This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such asStochastic Networks, Analytic Combinatorics, andQuantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By usingComplex Function Theory, BoundaryValue Problems, Riemann Surfaces, andGalois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize theTransient Behaviorof the walks, as well as to find explicit solutions to the one-dimensionalQuantum Three-Body Problem, or to tackle a new class ofIntegrable Systems. Part II borrows special case-studies from queueing theory (in particular, the famous problem ofJoining the Shorter of Two Queues) and enumerative combinatorics (Counting, Asymptotics).Researchers and graduate students should find this book very useful. | Random Walks in the Quarter Plane by Guy Fayolle, Paperback | Indigo Chapters