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Discrete harmonic functions in the three-quarter plane. (English) Zbl 1483.31036

Summary: 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 three-quarter 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.

MSC:

31C20 Discrete potential theory
05A15 Exact enumeration problems, generating functions
60C05 Combinatorial probability
60K25 Queueing theory (aspects of probability theory)
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