Random Walks in the Quarter-Plane

and History.- 1 Probabilistic Background.- 1.1 Markov Chains.- 1.2 Random Walks in a Quarter Plane.- 1.3 Functional Equations for the Invariant Measure.- 2 Foundations of the Analytic Approach.- 2.1 Fundamental Notions and Definitions.- 2.2 Restricting the Equation to an Algebraic Curve.- 2.3 The Algebraic Curve Q(x, y) = 0.- 2.4 Galois Automorphisms and the Group of the Random Walk.- 2.5 Reduction of the Main Equation to the Riemann Torus.- 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case.- 3.1 Lifting the Fundamental Equation onto the Universal Covering.- 3.2 Analytic Continuation.- 3.3 More about Uniformization.- 4 The Case of a Finite Group.- 4.1On the Conditions for H to be Finite.- 4.2 Rational Solutions.- 4.3 Algebraic Solution.- 4.4 Final Form of the General Solution.- 4.5 The Problem of the Poles and Examples.- 4.6 An Example of Algebraic Solution by Flatto and Hahn.- 4.7 Two Queues in Tandem.- 5 Solution in the Case of an Arbitrary Group.- 5.1 Informal Reduction to a Riemann-Hilbert-Carleman BVP.- 5.2 Introduction to BVP in the Complex Plane.- 5.3 Further Properties of the Branches Defined by Q(x, y)= 0.- 5.4 Index and Solution of the BVP (5.1.5).- 5.5 Complements.- 6 The Genus 0 Case.- 6.1 Properties of the Branches.- 6.2 Case 1: ?01 = ??1,0 = ??1,1 = 0.- 6.3 Case 3: ?11 = ?10 = ?01 = 0.- 6.4 Case 4: ??1,0 = ?0,?1 = ??1,?1= 0.- 6.5 Case 5: MZ= My= 0.- 7 Miscellanea.- 7.1 About Explicit Solutions.- 7.2 Asymptotics.- 7.3 Generalized Problems and Analytic Continuation.- 7.4 Outside Probability.- References.