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 as
Stochastic Networks, Analytic Combinatorics, and Quantum 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 using Complex Function Theory, Boundary Value Problems,
Riemann Surfaces, and 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 find explicit solutions to the one-dimensional
Quantum Three-Body Problem, or to tackle a new class of Integrable
Systems.
Part II borrows special case-studies from queueing theory (in
particular, the famous problem of Joining the Shorter of Two Queues)
and enumerative combinatorics (Counting, Asymptotics).
Researchers and graduate students should find this book very useful.