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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.
Promoting original mathematical methods to determine the invariant
measure of two-dimensional random walks in domains with boundaries,
the authors use Using Riemann surfaces and boundary value problems
to propose completely new approaches to solve functional equations
of two complex variables. These methods can also be employed to
characterize the transient behavior of random walks in the quarter
plane.
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.
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