Discrete harmonic functions in the threequarter plane
Abstract
In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane  resolution of a functional equation via boundary value problem using a conformal mapping  to the threequarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.08082
 Bibcode:
 2019arXiv190608082T
 Keywords:

 Mathematics  Probability