Random walks in the quarter plane, discrete harmonic functions and conformal mappings

Abstract

We propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function.

32 pages, 9 figures. With an appendix by Sandro Franceschi

References in corpus (3)

• Random walks in cones
• On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane
• Martin boundary of a killed random walk on ${\mathbb{Z}}_{+}^d$

Cited by in corpus (10)

• Random Walks in Cones: the Case of Nonzero Drift
• Enumeration of three-quadrant walks via invariants: some diagonally symmetric models
• Green's Functions with Oblique Neumann Boundary Conditions in the Quadrant
• Heun functions and diagonals of rational functions (unabridged version)
• $t$-Martin boundary of killed random walks in the quadrant
• Plane bipolar orientations and quadrant walks
• Constructing discrete harmonic functions in wedges
• Polyharmonic functions and random processes in cones
• Green function of a random walk in a cone
• Passage-times for partially-homogeneous reflected random walks on the quadrant