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.

#### About the authors

G. FAYOLLE: Engineer degree from École Centrale in 1967, Doctor-es-Sciences (Mathematics) from University of Paris 6, 1979. He joined INRIA in 1971. Research Director and team leader (1975-2008), now Emeritus. He has written about 100 papers in Analysis, Probability and Statistical Physics.

R. IASNOGORODSKI: Doctor-es-Sciences (Mathematics) from University of Paris 6, 1979. Associate Professor in the Department of Mathematics at the University of Orléans (France), 1977-2003. He has written about 30 papers in Analysis and Probability.

V.A. MALYSHEV: 1955-1961 student Moscow State University, 1967-nowadays Professor at Moscow State University, 1990-2005 Research Director at INRIA (France). He has written about 200 papers in Analysis, Probability and Mathematical Physics.