Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics
Abstract
This monograph aims to promote original mathematical methods to determine the invari-
ant measure of two-dimensional random walks in domains with boundaries. Such processes
are encountered in numerous applications and are of interest in several areas of mathemat-
ical research like Stochastic Networks, Analytic Combinatorics, Quantum Physics. This
second edition consists of two parts.
Part I is a revised upgrade of the rst edition (1999), with additional recent results on
the group of the random walk. The theoretical approach given therein has been developed
by the authors since the early 1970s. By using Complex Function Theory, Boundary
Value Problems, Riemann Surfaces, Galois Theory, completely new methods are proposed
for solving functional equations of two complex variables, which can also be applied to
characterize the Transient Behavior of the walks, as well as to nd explicit solution to
the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable
Systems.
Part II borrows specic case-studies from queueing theory (in particular the famous
problem of Joining the Shorter of Two Queues), and enumerative combinatorics (Counting,
Asymptotics).