## Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic CombinatoricsThis 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 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.
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### Contents

3 | |

9 | |

3 Analytic Continuation of the Unknown Functions in the Genus 1 Case | 36 |

4 The Case of a Finite Group | 55 |

5 Solution in the Case of an Arbitrary Group | 118 |

6 The Genus 0 Case | 155 |

7 Criterion for the Finiteness of the Group in the Genus 0 Case | 171 |

8 Miscellanea | 183 |

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### Common terms and phrases

algebraic curve algebraic function algebraic function Y(x analytic continuation arbitrary argument assume asymptotics automorphisms boundary branch points Chap coefficients CoGF complex plane compute consider contour Corollary corresponding defined denote domain elliptic function equivalent ergodicity conditions exactly exists fact Fayolle finite formula fractional linear transform functional equation Galois genus given group H group of order Hence holds holomorphic homogeneous integral equation Lemma mapping meromorphic functions Moreover notation obtained parameters periods wi poles polynomial positive probabilistic problem properties Proposition proved quarter plane queueing random walk rational function rational solution resp respect Riemann surface roots satisfies second degree Sect simple random walk singularities ſº Theorem unique unit circle unit disc universal covering values variable vector Weierstrass whole complex plane yields Yo(x zero